Theorem : Let a(n) be the number of ways of walking n steps, in the, 3, -dimensional Manhattan lattice with unit positive steps Always staying in the region 0 <= x[1] - x[2], 0 <= x[2] - x[3] The first, 150, terms of this sequence, starting at n=1 are [1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, 1697385471211, 4859761676391, 13933569346707, 40002464776083, 114988706524270, 330931069469828, 953467954114363, 2750016719520991, 7939655757745265, 22944749046030949, 66368199913921497, 192137918101841817, 556704809728838604, 1614282136160911722, 4684478925507420069, 13603677110519480289, 39532221379621112004, 114956499435014161638, 334496473194459009429, 973899740488107474693, 2837208756709314025578, 8270140811590103129028, 24119587499879368045581, 70380687801729972163737, 205473381836953330090977, 600161698382141668958313, 1753816895177229449263803, 5127391665653918424581931, 14996791899280244858336604, 43881711243248048262611670, 128453535912993825479057919, 376166554620363320971336899, 1101997131244113831001323618, 3229547920421385142120565580, 9468017265749942384739441267, 27766917351255946264000229811, 81459755507915876737297376646, 239056762740830735069669439852, 701774105036927170410592656651, 2060763101398061220299787957807, 6053261625552368838017538638577, 17785981695172350686294020499397, 52274487460035748810950928411209, 153681622703766437645990598724233, 451929928113276686826984901736388, 1329334277731700374912787442584082, 3911184337415864255099077969308357, 11510402374965653734436362305721089, 33882709435158403490429948661355518, 99762777233730236158474945885114348, 293804991106867190838370294149325217, 865461205861621792586606565768282577, 2549948950073051466077548390833960154, 7514646250637159480132421134685515996, 22150145406114764734833589779994282345, 65303054248346999524711654923215773701, 192564948449128362785882746541078077821, 567944426681696509718034692302003744197, 1675395722976475387857861526496400455935, 4943221572052274428484817274841589781103, 14587540897567180436019575590444202957764, 43055804394719442101962182766220627765254, 127103430617648266466982424978107271745123, 375281510930976756310181851730346874521559, 1108229819877900763405338193186744667723583, 3273209089476438052473101825635320104642103, 9669131152389329200998265687814683780583133, 28567321136213468215221364999058944720713501, 84414794291793480358891042199686850901302514, 249478578991224378680142561460010030467811580, 737415571391164350797051905752637361193303669, 2179989657182268706949944711706683431675573969, 6445526902441229646310051859066530999375612263, 19060008608601035820122512539133495685366451171, 56369902000490155161466142878589232802345778151, 166736186990532204812594206164411016589993382759, 493253027399423689823559267775695878392662135604, 1459371636273993893967616514973002909449910595158, 4318344244242469812948502806410363711951592224347, 12779763960737611807939007796892996356569096666335, 37825235742770534271327510185799474362588759112692, 111967696262665849903769091543433376818806152053282, 331478378609143895971053565694243287254152991197499, 981449894682881651766048471053798615807057357319435, 2906234450266204129212174643462658253439431024500886, 8606803606778108365401717262737377633765146862076508, 25491831573907963336583054018749373611049855273869763, 75510488025438261304679120183518573236529303339176631, 223696821203921358836246695367291099954466200770942903, 662762421725143382344788171386348329048337151787800671, 1963815645947208854505196881559285819544210551279398181, 5819525803140027392592573852991803679728817387028471541, 17247168558452770365891669854750102048613779757551589252, 51119976623096674448512995874546202822306806110141305034, 151532376009683684148363191660038033005499043269537471329, 449222501434823367055795240724415226254160564831704494301, 1331858858333138229849628068560869645463787219753888975051, 3949070654397778925261595026952331124940608019718030910727, 11710379870808883456653370892680311031011070706105889565779, 34728484868917923073081492759953402358847792308802666162259, 103000390612050119846963061710720593823186046635306629775046, 305512870901106552768820055686476263875553152372320928076076, 906269136562156220773088044844995547011445535121944413744427, 2688570046955065969206424074262654900346481527533320340651023, 7976667118523625572095516219630936701843899260442051417052585, 23667751445225102552254506374989570190795491048526939496080013, 70230770489482038793218581951613637871985132295122357292290873, 208416578951506492873776608904240971601730012438544169566767161, 618544564576745831928921127489913959279036448705514455836614660, 1835875157466920391190868869901005367957215562683658293468490034, 5449394385047236914359114525390680488840392762143010623581084597, 16176546693038283993453107245062830999187843972123136291819669617, 48023678002714528839639173674188643621861261600288800950995865454, 142579341227678474194396320902682115345711749044188565056903397532, 423339640260024739139628371555493557867734086988317745954947316209, 1257048372285879591439531066344267341284748944902652545424979161569, 3732891328330226228496763135577770463979025464164833887161628222234, 11085839791400214754745367329223029411788809059033475245696786565948, 32924663292263290326176310930460243399440176801530892614734824373945, 97791968157406749394924901075381707518323470767341721197980700152917, 290478334914612646997850040852668570806349498231935309732754323955141] Assuming that The connective constants for the number of steps with i mod, 3, for i from 1 to M are is, 27, as it should The estimated critical expoents for each of these moduli are, [-1.500000000, -1.500000000, -1.500000000] The estimated asymptotics are [.3134386575e-1*27^n/n^1.500000000*(1.+.1875000000/n+.5750867935e-1/n^2+.\ 2041291904e-1/n^3+.7395872799e-2/n^4+.2503851397e-2/n^5), .9403159726e-1*27^n/n ^1.500000000*(1.-.3125000000/n+.1095920114/n^2-.3655317332e-1/n^3+.1190425915e-\ 1/n^4-.3971414313e-2/n^5), .2820947918*27^n/n^1.500000000*(1.-.8124999978/n+.57\ 83418852/n^2-.3973334369/n^3+.2688205375/n^4-.1783741173/n^5)] Theorem : Let a(n) be the number of ways of walking n steps, in the, 3, -dimensional Manhattan lattice with unit positive steps Always staying in the region 0 <= 2 x[1] - x[2], 0 <= x[2] - x[3] The first, 150, terms of this sequence, starting at n=1 are [1, 2, 5, 12, 30, 79, 213, 579, 1589, 4418, 12391, 34956, 99111, 282290, 807598, 2318671, 6676896, 19277213, 55785360, 161796289, 470175446, 1368646521, 3990131604, 11648937764, 34053774493, 99669953078, 292033596269, 856508055073, 2514356040438, 7387585695697, 21723351075240, 63925157577917, 188241181915415, 554670966099268, 1635395456595308, 4824562459598164, 14240463486503610, 42053981694387823, 124249388224815033, 367263175929653293, 1086037046999606621, 3212811629704787452, 9508015082937326788, 28148155678488305974, 83360353906750270852, 246950983793650205131, 731806567982168880724, 2169250453273074726206, 6431984765527952341960, 19076459873267587999941, 56592972494295308985692, 167932228671782854414384, 498434144572435081185430, 1479719563895940898397659, 4393853326681601760578497, 13049750770647281325747465, 38765525279294159269245327, 115178787750844528741840586, 342278418602858915102966081, 1017335243709332744891892983, 3024288779506206770474231517, 8991957176035459071487332900, 26739569988570145220662035757, 79528244192117486827175391449, 236566416935039603443107488380, 703796741850861240308814479665, 2094120821358470511426916327024, 6231814844426303046396205769955, 18547431735889716027111480552214, 55208736437539075348668088772285, 164355734094988777094375165034973, 489342872529490078989079811819950, 1457107025792205132423142085565657, 4339282451086252522677757651404408, 12923833037198749561413694541954148, 38495530895243634797566351026868345, 114676257794087204027517554135449933, 341648622685711001052494577525842722, 1017952694690980993903090213539530317, 3033305758017341065849036042921307724, 9039498351609077821076189751107622877, 26940828117644895590881239341467868568, 80299898933804595101714095413419968949, 239362174627459127334535435094155437226, 713561829294927568999899194661893848467, 2127366645395136458016143962782722913002, 6342885131798423552694824269287839528761, 18913167531934513728291241943422008187508, 56399311887908480904927338034904624349355, 168195627779754571554400561445714249207414, 501633117707562447239382434738416056564009, 1496192831929161473897277157367125894640532, 4462909780830279738238441754491331923938869, 13313036983817707876867624353696199755428632, 39715865334844659766800419433625772240258052, 118489030321047143741524301175321095175560558, 353523976096615210521764036392799502302223994, 1054837650619315044309897565429432170870562071, 3147587594511225876261330188402984569268797697, 9392797118155661012310119030761754265375107546, 28030860935787725729503786091362983879914270554, 83656904599380751945952423637319461965750509107, 249683879788851955376180645572741486827586372436, 745250183169972236837799946288672563071413011018, 2224518863077783493452830372941283540470219518312, 6640366513915199682850965260676675649617083420299, 19823006467595883707723689660664308352786944027308, 59179062343960571653634716627067061417521227916351, 176679965928731837244273271485828054788806462916305, 527505275235508512630810625277705184867382307266228, 1575020416073317479539303325200472187959123565918924, 4702892669533679485167540063881831220889748683232505, 14043102845822154159269900255299521995193217550356889, 41935309866164767786868236840153252433161161742422504, 125231926880454556967549351406394595408646470463501620, 373997204360327641535544291993547676194610669369585805, 1116964588513537834088898576894657543894668081567295530, 3336014185899032962132752728879128963442821492308608515, 9963994882989165999243841741306097329758129534558030049, 29761576016721670273878384500605853765588480152174732643, 88898591737644281866518693720239603539589352738052464148, 265552304683850017369921401248406359185775705250229364923, 793270357328463112075541927656417428594819709031731655388, 2369780172346987462021331601347407638650437861635109687123, 7079626030572011252160069135809008943696098018038505568338, 21150845558704151066569651114242857903678246424304268148455, 63191702806500882206589472105588902414179355877609159660659, 188802196966131354496343870517895667491668846502391041131499, 564116041336021597732006215774943552505190964341231502045441, 1685559328468381243486827239216807456508594342485913550125498, 5036555176861081992607115303999401390972173762348386893035157, 15050013314577929639851809320558658093841628327128872779409749, 44973189581175203327841784909477325061487453857088071159127963, 134395213001445158322959106703017497505139561686521674648542932, 401630798382787858642267291013611482167938537972340222008879421, 1200281500359564128578643632736980575083464058893349839126730415, 3587169624857915263732405403997646492361812009626384445891290509, 10720948700578433536407103704206244692462077442576276545191807856, 32042532750450858757365178786804456351582684079904914784483484487, 95770683360678711290463729044077181663672453984336649692596588891, 286253157200048491092804687689087939820995557560364335531763860444, 855617694625070766651340613411793279633774786465620235707071198005, 2557530547822024440862209252400514032072390912492837983077740920949, 7644926836177829125846767919070163687351806032622489408140332674522, 22852678190387081512628397365883075327064138538874716763782861315215, 68314356152442216900681080205080284297336706969041263538357361903618, 204219779061442825927867989930619735532035575896145333490538850522652, 610512297815760424995111718256812218753458155644074076057914011315376, 1825163065754109254483912261883113389271353479579025070676772673252124, 5456566089739613924040624950809241902140916339687062808720726203772521] Assuming that The connective constants for the number of steps with i mod, 4, for i from 1 to M are is, 81, as it should The estimated critical expoents for each of these moduli are, [-0.4512974081, -0.6025048236, -0.5149470208, -0.5031268982] The estimated asymptotics are [.2471424493e-2*81^n/n^.4512974081*(1.+9.689703825/n-258.8051236/n^2+4357.30949\ 0/n^3+8506.003466/n^4-1831637.849/n^5), .1794427937e-1*81^n/n^.6025048236*(1.-\ 17.06443520/n+776.8089951/n^2-26402.97197/n^3+743057.0880/n^4-19877018.61/n^5), .3224589486e-1*81^n/n^.5149470208*(1.-1.654978459/n+81.97422463/n^2-1955.423163 /n^3+30154.82898/n^4-325331.4321/n^5), .9034983114e-1*81^n/n^.5031268982*(1.+.1\ 693357390/n+16.13971785/n^2-403.5210382/n^3+6310.770063/n^4-44159.08988/n^5)] Theorem : Let a(n) be the number of ways of walking n steps, in the, 3, -dimensional Manhattan lattice with unit positive steps Always staying in the region 0 <= 2 x[1] - 2 x[2], 0 <= 2 x[2] - x[3] The first, 150, terms of this sequence, starting at n=1 are [1, 2, 4, 10, 25, 65, 170, 459, 1253, 3465, 9628, 27020, 76248, 216477, 616633, 1764643, 5065781, 14590983, 42119908, 121909278, 353549512, 1027434657, 2990386213, 8718266640, 25452116682, 74406622612, 217761251813, 638042779002, 1871303118628, 5493621881839, 16141036068951, 47464129306441, 139674615728911, 411320504291137, 1212040983060388, 3573766200698325, 10543381799320431, 31122308389380092, 91913367277273297, 271579240875232819, 802801166061222967, 2374142690309455706, 7023892922288309802, 20788201249146791647, 61547730918677606919, 182287991890607996161, 540063185484073879493, 1600543921572537572269, 4744809448035776214506, 14069988655317563560883, 41733531686917411459019, 123819609999800435608741, 367451946525413700915306, 1090724023821533108632365, 3238374039151754544661178, 9616873887341781955239728, 28564792228934730106274166, 84862428169339626578447130, 252163954925609059633091280, 749430150121961573595934093, 2227698604837016403687083951, 6623024019625211239076927295, 19693718099606984394007393897, 58569082538726911777509861926, 174211057166299186228015797627, 518259800931489056184932559971, 1541989357293332959595646968604, 4588549652016279961064138120740, 13656126415189690655130991436332, 40647694009173405403796237996658, 121003688998558022295157746894179, 360258357613779577145849361219057, 1072705516723700274553622438262422, 3194452906146590959793041035677019, 9513939446050191609156057051372376, 28338109068665319832206573947663639, 84416335780745777119565256295042933, 251493072102882323686679063609525450, 749321383676918890202289776249956162, 2232809471003673236632533398375299262, 6653887291473675265021161337861327150, 19830715663719751524432454557720363000, 59107065526121340074955109785887745566, 176188476723940848056172103473254298008, 525232609631279017829548001545116644389, 1565889205215162295435852831497446277596, 4668792894773867258154299940137869975306, 13921357301040984985843456035087500497939, 41513665603660998758977261897912048460014, 123803331179355466117762514114731065178572, 369236414417919071904955320436653522134676, 1101303295499979143579875592463734069147042, 3285026180552838189494063392122919187650183, 9799401029007295169074012833368927225300520, 29234006585833797665261870286083505795729855, 87217699896241938869518641689633799868272327, 260224247386892678285031470458534003977843456, 776456518514597090519544481790647573902536033, 2316926049294117644844473288295413078080701619, 6914046595059771343145071682442866958401120538, 20633693530340082964763862797039572385121800672, 61580848846909021574644161719230492132938475743, 183796767718782806656128483635140517186315114653, 548596563166503261410330483636827111037271302269, 1637535883325718684891659888033789956667372202141, 4888218810160892149382825800235607503312768181342, 14592579391972138628554531078736143637684153187416, 43564694631800701558328532674297583409945048828612, 130064279465942395035742076954967295679365514415432, 388330724167314644937630421173585783609819251084433, 1159485741582410507341217618320646382282376435337084, 3462171962574234908944227954869806041474642424280996, 10338345522447600469936958573161218801937794217768616, 30872533686247879797453450547325845447024095150684370, 92195976742802338473187548342764082483143358205317924, 275340305773713773042215856755860053464604378820490496, 822328607914563581255360299240756616885757121064353008, 2456057172857985551317466556368648601368522715190689836, 7335820428118020363454964004517436649136881158214311288, 21911683781230190013835464805760740978286610674537837531, 65451461970653863677219876893850365692506509156555780975, 195514603407234435244854253677680491173717128226722043569, 584056707674674099168493102357454659525363369188332802894, 1744803487788430612428296503889127767108458491963242702752, 5212588550541728575779910070757594485572679494981930566795, 15573116408193225730382370001467609533971345183713306904217, 46527799916187510099863441622901058351565497172686048264116, 139015798851722142935524008399144653104306818890634974217965, 415365301861220595338336564118707321854001359449091528271686, 1241110486468447833704594503361519973441584093103680136283678, 3708554118561843329750876229576723143644505443355904540359488, 11081856385919711508016334894470323457737370497462829687529668, 33115698144420428308942678094401204120081699050582334787933189, 98962031064270419988964355671553627016633017104518824553985437, 295744286068797851980100446656818126994553147692487355967569225, 883846811016154584365762377314473832868729421826677917804257094, 2641498148723109485302854148442659974311127302411029288110677350, 7894708561128414820658347582615459805167448064085935735891115135, 23595771620771894629134206671568948319152441700459436957550711792, 70525207137409583875310942966560667055356265513180349275981496530, 210797996892458716430418312990639177493152502405779043307017500261, 630086732291683883945530354042012793969686968742161346084097274593, 1883413860549883434738036414875453992938910805494905106471153456835, 5629924782287305054500661035609710110297123682641869506936026164033, 16829478557852921630645793973810468447058288531945878401990867266983, 50309477016316383819202956282967254434238482355469651683029289735596, 150397238405510646853584882146133170170719146816762467240840224897096, 449614886827434109962310417974490062050063539186976744895728398621440, 1344163544154078473374515202718144651594525539972275407235436259002863, 4018592817151725871686421939653498878072992046171104541810755402087472] Assuming that The connective constants for the number of steps with i mod, 5, for i from 1 to M are is, 243, as it should The estimated critical expoents for each of these moduli are, [-0.4668690829, -0.5204045726, -0.4737056073, -0.5183056192, -0.4759496325] The estimated asymptotics are [.6005396467e-3*243^n/n^.4668690829*(1.+5.418000334/n-107.8401794/n^2+1534.5873\ 17/n^3-3502.009311/n^4-223278.7075/n^5), .2429084696e-2*243^n/n^.5204045726*(1. -1.924326842/n+74.84297847/n^2-1488.581687/n^3+19908.45877/n^4-199584.9286/n^5) , .5615400203e-2*243^n/n^.4737056073*(1.+4.286803165/n-88.41020788/n^2+1314.845\ 071/n^3-5260.779007/n^4-135603.4880/n^5), .2160699595e-1*243^n/n^.5183056192*(1\ .-1.838531468/n+66.53238610/n^2-1319.217438/n^3+17331.30921/n^4-165828.2160/n^5 ), .5117514656e-1*243^n/n^.4759496325*(1.+3.785309500/n-82.66518179/n^2+1251.01\ 2695/n^3-5827.973653/n^4-111072.6684/n^5)] Theorem : Let a(n) be the number of ways of walking n steps, in the, 3, -dimensional Manhattan lattice with unit positive steps Always staying in the region 0 <= 3 x[1] - x[2], 0 <= x[2] - x[3] The first, 150, terms of this sequence, starting at n=1 are [1, 2, 5, 13, 34, 90, 244, 676, 1897, 5355, 15173, 43208, 123696, 355732, 1026527, 2969894, 8611239, 25019137, 72835240, 212416299, 620454375, 1814760582, 5314389277, 15579957694, 45721391626, 134302766240, 394847366312, 1161770246656, 3420814337962, 10079359755316, 29717333880264, 87668121036623, 258769865098487, 764206371506319, 2257970405842603, 6674570254187449, 19738495279174257, 58395560470855158, 172826769124850523, 511680960269902214, 1515431855917586671, 4489673800794255394, 13305378150184466610, 39442744684555745931, 116957599153768822239, 346901002331607932399, 1029184884788598285874, 3054127110001064068195, 9065297875899139646267, 26913736008327708493001, 79920611361201672480347, 237373772811339420723187, 705167442998238846011728, 2095241489138367166700308, 6226659179996449066525312, 18507693961957464299464115, 55020295104202507825702764, 163592827177309733377737954, 486489901180180266147698271, 1446935782516189663246767096, 4304159038112375699277708683, 12805273395751381500117945386, 38102095305765932097791596132, 113387814896513567092375134955, 337473438161176806873773618134, 1004538851137036556225517329738, 2990516090020588694733531056914, 8903817056960067878265999986523, 26512792543103969284084971699933, 78955507005204801450228466866655, 235155801407416926990015293079814, 700444866616409151554217507388802, 2086584287934585092566695137471992, 6216420835521797850587111217397332, 18521929030996661137909450120546179, 55191505213280208225451129782404219, 164474055220200478219860405672288857, 490185734957248657492489771950848723, 1461036450686546409491144856835145462, 4355094614544633409039308339257012681, 12982830347435881557058583333747614349, 38705754029901344692587817531874910623, 115402494476868023631532615182839076983, 344102284249628865370940331222602375068, 1026105041117705041887571541041437962102, 3060040911883646779587054075046504743964, 9126264621211978068167924684614480761222, 27220029187155821867554813520118032683858, 81191981048559456045271262261080111079376, 242195490270886563767262162349903433021552, 722514706295191054712463181593932325772602, 2155531823424963958358911324336936844545567, 6431150892315608506068101857384404032978810, 19188847713540935971003580516887457167210480, 57257775526880853037475003599779450273506140, 170861764036632938549452690948134779558375167, 509893664889613358332361300769302711038453198, 1521731939301732399010206810299034696862675583, 4541716428083442537828028370499608575488912810, 13555786373646100086063618382773092094501999561, 40462414927961954954110198093492649294922415505, 120781610621655980055751033754077174623444516269, 360554843488419427124131797123889865896924052516, 1076373334051270379616029539238577742628785805772, 3213476998795705649362231185595450426045829372721, 9594177159195360264339941007113812711992854341384, 28645747385345442218305510326583588063253208653744, 85532682593988170519336941261094971756615769049171, 255401342272745468561481674558097849937350596528385, 762663654253417092268011080235138955043037895848119, 2277515663085168650910383731267992593737408531969040, 6801548627630057671711753282801695107860777635821008, 20312904595881010893974601259544088444747863371571469, 60667167572363843291028382773242831748023002941026511, 181197653529067768844362911866692651031411300228120338, 541213073331226437082834116223316323727577191448722475, 1616592479496526781741907545058044753224352396278051328, 4828909987708628848558117238822954733547150864032111265, 14424928480025089391126880529538861660805833453931304509, 43091732818054733284047713157606815014041890268471028115, 128732948198720363871057771133562060596318171162678303810, 384592385956553838319235603903014185024991301738437588833, 1149017403502530574009426967969758583777857226639130666399, 3432948155360483398480950977976843461438651202799279703948, 10257047265700958832137598513295496348278096891708981436962, 30647264015041228227636203487086789651451086454268462588150, 91574609531958967191436927475038554909379567904981275840862, 273635369217298145957039466120883169673760248869697639163647, 817679222788917086356814071458744724244975372874813144274967, 2443470243083398567308360359748229721034166302779114234569030, 7302041572577969166650137044173803866264033878805894943463074, 21821997428260211401190833493001285788942031305714244637180088, 65216494960481369896035777279323991439189248380185196871419223, 194909482992483296891092466826069997256347204339663462914025688, 582533435799875129255021568203078029218376385980203557466926939, 1741088849400704862742846743207888371344949935558961479530286638, 5203948839336158618466116620708004698032534025745547129779119135, 15554531462692372726031867379222976216214159600152213615966638850, 46493528575427407481671698269727421704337859404247853853064059571, 138975924413134554156058066708261044956818368326751112478220142797, 415430099552416171359556048114139339639447705385311823094066436332, 1241845336299094352918712460042082860085738424952636970425978774139, 3712342746569970079968536174564186214689669124272709329671936456847, 11097865861144144983292188951054187398453215759678782290113501714043, 33177340822583036531827079397147325182604205291551150703673947822058, 99186879348387620345159841333256915222043003532793764102848835530023, 296535902914579311400817957181836781042010148980041420334628050186457, 886565037688045019656348642667885992423614002623330619777903697874389, 2650660096953983889787581565094221984373343384910647701747644809108391, 7925149529554954884421590274992030843940036173133193155101843959136483] Assuming that The connective constants for the number of steps with i mod, 5, for i from 1 to M are is, 243, as it should The estimated critical expoents for each of these moduli are, [-0.4999969133, -0.4999498238, -0.4998912795, -0.4999240212, -0.5000878345] The estimated asymptotics are [.1440674728e-2*243^n/n^.4999969133*(1.+.5635053246/n+.3385166712/n^2+1.4688839\ 94/n^3-24.12596477/n^4+192.5680324/n^5), .4320883031e-2*243^n/n^.4999498238*(1. +.4700236956/n+.8731087625e-2/n^2+4.476085921/n^3-55.08320009/n^4+311.9361571/n ^5), .1295838785e-1*243^n/n^.4998912795*(1.+.3782028149/n-.3399038015/n^2+8.500\ 749491/n^3-96.33942492/n^4+478.7926348/n^5), .3888224838e-1*243^n/n^.4999240212 *(1.+.2738355262/n-.3193805690/n^2+6.438851405/n^3-75.91346469/n^4+396.0783285/ n^5), .1167535613*243^n/n^.5000878345*(1.+.1515397920/n+.2189582785/n^2-4.43555\ 1865/n^3+34.46674437/n^4-71.73675028/n^5)] Theorem : Let a(n) be the number of ways of walking n steps, in the, 3, -dimensional Manhattan lattice with unit positive steps Always staying in the region 0 <= 3 x[1] - 2 x[2], 0 <= 2 x[2] - x[3] The first, 150, terms of this sequence, starting at n=1 are [1, 2, 4, 10, 27, 74, 204, 568, 1597, 4529, 12990, 37579, 109346, 318904, 931817, 2727628, 8003531, 23543327, 69418375, 205065889, 606543183, 1795799637, 5321461283, 15783506546, 46856411862, 139223725542, 413983076887, 1231758882096, 3666905356226, 10921529501568, 32543754405260, 97015609436159, 289330798229075, 863197382291679, 2576156025793708, 7690687939646773, 22965658933900994, 68596960648268934, 204944671467963970, 612445492188782439, 1830587645535904283, 5472667403015103396, 16363880057294611617, 48938055390630885524, 146377880953220760853, 437893503928100028361, 1310152665057181775722, 3920415006420544468779, 11732627694844355309968, 35116263631189434653114, 105115796282729930173997, 314681779906938552751572, 942142642530109568824128, 2820985305968273974362145, 8447377773148116197220119, 25297514584956759855322870, 75764665762561652313941098, 226927257870687969870593027, 679729403963418668404396833, 2036166709353810056713064810, 6099820430617973822216715937, 18274514613613210525518810653, 54751807178336141784429083912, 164049033001120377013625675844, 491553016718298752495335821867, 1472948205515179651867241535213, 4413915454255792671684604669760, 13227538698683656558476398110022, 39641646818245525827016036986612, 118806752863701283457251534155790, 356079173615579761235679341853378, 1067252774068140256669029226214350, 3198914162783302601747366137384167, 9588524897528042012159320415105178, 28741823186307793856050364781529162, 86156795701371721573473727382798101, 258271759198408551812476029840315770, 774240585442501410576016835196861198, 2321058451758630166658276247119981172, 6958359789351729589495910721136408254, 20861133125198186656632697936294817704, 62542997754877744641257704370241346691, 187511917124763233128946012682352156169, 562196389288939180985211142418376656890, 1685605201137249279894772571906603435280, 5053961847916411423286986707807811497024, 15153606692888994438372892541195403533609, 45436796528827811046191645589844297178933, 136240660210283012820858763562141242683108, 408519537529488803008199406612106654433144, 1224970721797143547165418787733043114503296, 3673204533886817397150248689394044809182383, 11014652325188718848262378665160487472168698, 33029539403155816845988306074623750740351206, 99046711051583058339306816121058220188840845, 297018296084418460339620143091094977482178093, 890700594263831742815065153722563498698885981, 2671071303238370450126906661810390127758593924, 8010216099005751797969846619594671945624112168, 24021925485283944858355705305852143231543335563, 72040390451118855752768874901263739412140754091, 216047276174139402782426519196164224885186458561, 647926689123969741204261500430756680860409972705, 1943153591595904413755887778220185771024362330506, 5827636175685568478159790703071412861758181537857, 17477593470095084048025257565312960964016536841806, 52417294949513680307750670355588489260277020358053, 157206760121668506572749570870634141674784161443364, 471488764582373299330921624036051460971925025216278, 1414082928668438302758158557310666714732180333666699, 4241131106560067913385495198862672298853889859384991, 12720134272481148955043798192367504744897772831464627, 38150898270305133005912306052684168491576333697837646, 114424971933502765131650671836977737912495939850633129, 343194041589554831701844864656447696347936804089610073, 1029346160921468112459940285684231108169542650513939514, 3087349920260437827553680542999393980128254039279819186, 9260040193250959740066852113339472710971241878113761034, 27774254849963814256600730432398757699980408908418257986, 83305640730744780591472194646260946167372182659259081848, 249866925960927292142776482881932036409760192527732052999, 749454785253600629591900352619754189186421785298822150540, 2247937991696319617131654362908491862830978565255994061653, 6742568642717351520901226965747347358946074363614214800845, 20224068084689347926105824429172226979183522096343266833418, 60661576148856186019885832480328462024082854603041405278821, 181953674280830361985698687834424177975028548928470750052802, 545770275237043895029608767086617218081642270207683211563765, 1637045608998688963490209622329697622312788655059961110100546, 4910361622742812632618183723613503672083754952822204001755298, 14728818762766909032416437191185458498090868578092541885543692, 44179831195333903554629298857126968322371885274318279032668352, 132520122635712082175734658072360953308352312214460740581401843, 397503723611387291513221024095283446735656661155663422288494132, 1192345514907105960241254395151321297296256155974236616784811034, 3576552034917246454730161501717069879750688724566150243088725288, 10728238868597975056611889331545606129631192908047597368350422164, 32180570644690827567525780447970738862420269008511680070593474882, 96529582211365885895441335829881795962587698899012178624857190507, 289553255613630611959394037698835458594913507719491733315323215325, 868555911844213095929484262862338414850743893174929658879360579673, 2605363803168748007320051973434536972026019542168765206488171108925, 7815201867418925573354017226745776893409896658432774718083335977440, 23443001875359363699468265414032768957558114196858989464084657610039, 70321383728294671102043769778907894739815886155827826629819727200400, 210941837619296206829457487512084347479084778263252675937323235541437, 632760182896361813049468205758022688222040768263500672536326954045992, 1898089258440282226329960527014097168290451731693418628644532332834081, 5693707618408298674060125000002398406184184118280337469640979668223137, 17079482407276263832924979450132294443777218156680405985223159654007313] Assuming that The connective constants for the number of steps with i mod, 6, for i from 1 to M are is, 729, as it should The estimated critical expoents for each of these moduli are, [-0.009800604700, -0.009678525463, -0.009551294235, -0.009441700854, -0.009335337218, -0.009230663113] The estimated asymptotics are [.1998544847e-3*729^n/n^.9800604700e-2*(1.-1.310657408/n+28.88690435/n^2-251.07\ 97554/n^3+1823.705070/n^4-11024.29615/n^5), .5991575048e-3*729^n/n^.9678525463e\ -2*(1.-1.296061573/n+28.63668187/n^2-250.8560543/n^3+1811.821202/n^4-10929.9282\ 3/n^5), .1796205640e-2*729^n/n^.9551294235e-2*(1.-1.280875435/n+28.37111270/n^2 -250.3540305/n^3+1798.124936/n^4-10819.60325/n^5), .5385329969e-2*729^n/n^.\ 9441700854e-2*(1.-1.267620748/n+28.14907986/n^2-250.5321982/n^3+1792.546823/n^4 -10767.03676/n^5), .1614641766e-1*729^n/n^.9335337218e-2*(1.-1.254719260/n+27.9\ 3400610/n^2-250.7967580/n^3+1788.888613/n^4-10726.53138/n^5), .4841099194e-1* 729^n/n^.9230663113e-2*(1.-1.241982964/n+27.72105731/n^2-251.0609076/n^3+1786.1\ 48543/n^4-10690.85258/n^5)] Theorem : Let a(n) be the number of ways of walking n steps, in the, 3, -dimensional Manhattan lattice with unit positive steps Always staying in the region 0 <= 3 x[1] - 2 x[2], 0 <= 2 x[2] - 2 x[3] The first, 150, terms of this sequence, starting at n=1 are [1, 2, 4, 9, 23, 60, 157, 415, 1106, 2994, 8247, 23003, 64560, 181541, 511242, 1445460, 4107772, 11734683, 33659461, 96795325, 278779877, 803912190, 2321930924, 6721321968, 19499432399, 56678316264, 164983629179, 480766553225, 1402261671168, 4093846287245, 11964229551170, 35005896366086, 102532166783834, 300591655695426, 881914635331012, 2589204822848713, 7606426784887577, 22359840597952061, 65771128073490408, 193593211538793002, 570177290326283374, 1680226607599227298, 4953813007279556851, 14611970141617593793, 43118833492944571088, 127294565990720911049, 375954338347624486396, 1110822654319565097847, 3283423699537777366940, 9708882093261777732351, 28718387967112867176140, 84974908747675444381221, 251510129906003921539938, 744646820215643098287234, 2205325422231897273089629, 6533153954230782800255654, 19359514481676073923106161, 57382648769699851990893393, 170127578228072370427769808, 504511996594721172520691549, 1496466592315199888094946670, 4439747993050345692357065618, 13174779188759972382836108985, 39104069328992742042592871329, 116088763210462696754462973200, 344702061367511363554782262347, 1023714700107672046500077566841, 3040828125061665709715983874619, 9033998127035603391213374467137, 26843625374683903878667104899768, 79776294237184177190138291190051, 237125202285187959098710980228764, 704936720507896405584179812071327, 2095986066496126501421856131508360, 6232894410725034643619087757118797, 18537531459006112585383852948067027, 55140778429669244285805366561535455, 164040540054702054127387589407566951, 488073945038675768004526164531812168, 1452365028293748099904755291081941593, 4322349365749727154336404469855784320, 12865183971041379038301026663332222893, 38296784305161200407691409968891659083, 114013722528826165836455039004243428951, 339468187039783718167262062481451964829, 1010850706953683629235234067406260126823, 3010371087589700411522594831964373811392, 8965979020418211783633773844264424062290, 26706610938458738458123699415328227221533, 79557637793986256142557345861891121314287, 237020373411779243438688975819324529212975, 706202039861108030827772360378623322414767, 2104314901512023652652156524544486516923468, 6270903078536142779218394812146007064368950, 18689011428954709422657347334873190955712867, 55703062999657329170303719217340046325021805, 166037954653286352864365326321197449216765559, 494960155388569692192165552743914551803917826, 1475593333869603209340991379926178853561962235, 4399423923671110417268431784700131037028252642, 13117674438120854188030685478358643793138145269, 39115520392660552100394893346513100038819067449, 116646587631455427741521332372200948288824184148, 347876657788637457299213240523486540172014235810, 1037547861891954374735006552433968262297513352761, 3094709218401681783001683778317582246952402557466, 9231232140203745901803046817129572197156951083442, 27537653250293568066365472067113706398323213939851, 82152543238545673035182846067594848839705358658209, 245098917778342517162251310946164854881622909101949, 731286596331506560623255416364609089711346474109018, 2182023257041005551203905294903073353994678985994961, 6511126285057127245933871145835970889539332339190698, 19430200040917369622245367883643355866246680257922215, 57985881361546483989820057599105249350347001833125787, 173057553424375461885008022909224935435618971604852236, 516513515146623232645231790098315842728295624386278292, 1541683502687500700196977258739920817105448255306452724, 4601832356460886977843267945437072745552337195703473141, 13736880995464296815586569781649974231265382375412056183, 41007832124432787520464509167343233395537748786186329777, 122423944829856602807956098822774841059900484296968673911, 365499143064034835168846186091574848221238002791017088408, 1091255314095734663035331918776922005554456391829148779504, 3258262097634764616761721979832454636874413768171909106677, 9728927139983255152495394263814346528892038251284150175457, 29051119228461641355536120669998766198400529314162730138617, 86752009785087331799863714876735620345129980576548410279931, 259068509033145522564202344289192216691910711029934644946848, 773691370714191498202194874427483709918153570313705407558853, 2310673423895595517371412687563788711825692481812455662549853, 6901233753994179073570832513718263122808776151933642441582250, 20612560864343438286281383635143392602084650887187933276289713, 61567841094662531264217751529086606260359226045006805636202608, 183904520792214587559714856203447001753977578509770430584574546, 549347555436034892864864461364531730089895376123048535367187460, 1641035631402744059976433501985732570431722829038048958257888941, 4902352766986244555981371621751302284897350726913957998131257018, 14645579227585218308552762919241963180943182865501408340715632454, 43754599875491888173403010905189764520859982054205821751313274435, 130724141797139001538084348005998546754276009693789946073785630989, 390573312880685645461767969003023635698990788270066088707808018044, 1166981043485151583662755648410994139948573365636895387259680495349, 3486898753825337843791417509174979499360324724421683868746089376439, 10419069555160322713676753074966733053457629825624664026749350222076, 31133816671425475452330944951852044215854554112500240277778120766039, 93035646571735935616715395509133100291062302346037646706574789645455, 278022382789453855448778752574982327409476828754276584694550690132787, 830851287426620738865949872355106189823996290865285079457939718434534, 2483017564194431349428759699048697220759056862730249520701705961041046] Assuming that The connective constants for the number of steps with i mod, 7, for i from 1 to M are is, 2187, as it should The estimated critical expoents for each of these moduli are, [-0.2324529943, -0.5977241669, -0.4951931489, -0.5255723829, -0.5017489070, -0.4861944638, -0.6976022861] The estimated asymptotics are [.9961587433e-5*2187^n/n^.2324529943*(1.+26.29718363/n-109.8629492/n^2-3144.603\ 806/n^3+77666.76005/n^4-144392.2737/n^5), .2006597571e-3*2187^n/n^.5977241669*( 1.-7.932967228/n+213.0015793/n^2-3630.818026/n^3+52996.57758/n^4-745272.8917/n^ 5), .3528150590e-3*2187^n/n^.4951931489*(1.+1.566670134/n+4.190524940/n^2+106.8\ 139590/n^3-1286.176434/n^4+3918.905401/n^5), .1239795637e-2*2187^n/n^.525572382\ 9*(1.-1.321751977/n+55.72115256/n^2-613.8532840/n^3+5240.865408/n^4-43265.33101 /n^5), .3288044927e-2*2187^n/n^.5017489070*(1.+.7502415147/n+16.47965573/n^2-72\ .15820938/n^3-12.52363413/n^4+227.1923061/n^5), .9091671411e-2*2187^n/n^.486194\ 4638*(1.+2.177020292/n-10.36136685/n^2+254.1183663/n^3-2034.579420/n^4+975.0547\ 562/n^5), .8177165445e-1*2187^n/n^.6976022861*(1.-17.27973042/n+491.0937779/n^2 -10804.26323/n^3+209717.8171/n^4-3838019.173/n^5)] Theorem : Let a(n) be the number of ways of walking n steps, in the, 3, -dimensional Manhattan lattice with unit positive steps Always staying in the region 0 <= 3 x[1] - 3 x[2], 0 <= 3 x[2] - x[3] The first, 150, terms of this sequence, starting at n=1 are [1, 2, 4, 10, 26, 70, 190, 519, 1437, 4022, 11371, 32351, 92429, 265110, 762991, 2203387, 6381111, 18523685, 53882646, 157014481, 458300067, 1339688171, 3921353792, 11491855210, 33714128297, 99006797325, 291012650387, 856093224267, 2520370503803, 7425320903588, 21890313791627, 64573437688629, 190591251026774, 562837287478460, 1662947388759324, 4915602093850149, 14536674892079698, 43006335465336373, 127282473879874828, 376846212585169244, 1116119145628330969, 3306738437686459905, 9799969134053475624, 29052157715362906346, 86149868142019280066, 255533352241903366787, 758144344636926861459, 2249896338716590807116, 6678435343288119298179, 19828239570147472556393, 58882563346319393710906, 174895456104527204555621, 519584282393660560294843, 1543887863720888777278606, 4588329298751176583718629, 13638590195324601368474104, 40546896708365574909353975, 120563632741021411387254215, 358544499892048477859753505, 1066437590191813200685246130, 3172423799679157339838015771, 9438612962971442064373159732, 28085641010957125919807604599, 83582960273210680206515337909, 248774899276531557499259220238, 740540846559615187251248631209, 2204669302622892084351323192991, 6564297828138044250608348945304, 19547083355399153975060357713841, 58213413720275913493488702715097, 173384471672514964825878524761685, 516466354899418277027481823417817, 1538570161236267249948352917597040, 4583896442407005413234324432996265, 13658196220866837212026607423564444, 40699759274192474454014769594855815, 121291175792855626503311452417055681, 361496766949766662857438201339544248, 1077498039000834234614158018137767707, 3211918160586941389419044704704701370, 9575191175751302865664401170197664507, 28547268513075532236139292366238072452, 85116732978562285158563465665783417435, 253803612039426803450035421085291228588, 756854391330698908528128426742257095563, 2257136380218681686742804813801676056839, 6731835074717500338352401162704352886400, 20078840654273216540022209154431546744532, 59892524578516919001525405257677998702342, 178663061931023535772899915555277544608321, 532996625482887921664155862663638633846963, 1590160893772339002290628415772746562966835, 4744429633814910722081919510895183841563003, 14156396266843307349514510483569732132901680, 42242208804273758430028789593143746958106768, 126056486622402874417692567550058916823114617, 376190592875798025057595969904030241258630479, 1122727382666679222110718640426683566823683958, 3350918483437680473499061280250057423374639486, 10001752508896783449524626120664504105291133568, 29854553475283454277518360863895027714823578217, 89118290652360920808605123107780955233671060331, 266038491777503264118243139737189558569364395926, 794224079290934348964675898607512662552126734479, 2371167101006099286862076175836120849500493313601, 7079480920114883645058357864148430799614715703527, 21137831524309839418789932162666197920557723377347, 63115908561447781148099944970842169538278588694668, 188467414587815366742631701680159348254757236199347, 562797791503917390107794146061365737545178485881754, 1680686963944090497582741824198156159193516060012447, 5019255245435870573069242822947405496284350537349865, 14990268476196190659962934233535586713327861383210091, 44771011422208643860288122342488917100804630832084588, 133721567446792004694523691116841653798144345061089535, 399413529443579595554532457372679733912739799469500412, 1193055245307888298780782245148134504433669161761364639, 3563809848587228374226067420498478410744194017423497255, 10645949544888037267427662549586639645873842326940642800, 31803134453837870143727103888400786110171332863917895585, 95010330063440253335065598618628004019546564428241084458, 283848662791944709025256201021207041521845060957943640519, 848042765184742565385450261419552285256072091992280544415, 2533747226225675843699399577901380991680346915509431303969, 7570476468566900112143524888459518039391468129765546908463, 22620245018078169537527071619211914357260490574590922247815, 67590449268935253680153830941951526105327119926711325592763, 201970099035499328281552863921714667425388774421903592931288, 603534811829832636882672702242150829019210661560101135452100, 1803561114731663612487253333135217817418002895758105347001453, 5389797976542187971041206451243586967645048989066016015946342, 16107457565499870887546002816654553399167702937655067505334324, 48138681806094700569154688655548487955606593163264107782241770, 143871207556998157361186116149210627063836103148307557995923657, 429997463543298901323538560348781850219545155540282309418387770, 1285197953992876134880638721665282317041483789969085468849466591, 3841369685615037315028054628589400404055767164362330148331447479, 11481905192026714370876007862611887717207047731539586357880949531, 34320484431895886754676957658247892682348078182595643439030405210, 102589826521060737638389602365932676092956681945476395594182181947, 306666580751538007963908541803402797684342973110151038860613027835, 916726351283707202252648565350628485163455412442684467768317463758, 2740462927492912480686070124867621245190845848353085586954499170343, 8192547071210533151192479343104275214228447915228130924579091838861, 24492020067699984511664710991873408628669896601603303036066212521279, 73221861845013341960337793321088454026394800137302185854929935844736, 218910848428550565402590584847726367302140228711196198520886335114626, 654491505885289189292583781136679636333354624268941731684455664705752, 1956819737850042903268589145945815157974218796021435748732029565001812, 5850696440406802359133812611728096173024453537661492535374206873594214] Assuming that The connective constants for the number of steps with i mod, 7, for i from 1 to M are is, 2187, as it should The estimated critical expoents for each of these moduli are, [-0.5001368669, -0.5001269985, -0.5001177011, -0.5002092648, -0.5001922667, -0.5001765832, -0.5001621195] The estimated asymptotics are [.1000601726e-3*2187^n/n^.5001368669*(1.+.5092750420/n+.5386246544/n^2-1.282443\ 546/n^3+.8812441351e-2/n^4+55.98917490/n^5), .3001652735e-3*2187^n/n^.500126998\ 5*(1.+.4386722647/n+.4240782929/n^2-1.346855562/n^3+.7046658265/n^4+48.68325870 /n^5), .9004526773e-3*2187^n/n^.5001177011*(1.+.3680293580/n+.3250154646/n^2-1.\ 369839810/n^3+1.292411716/n^4+42.25855889/n^5), .2702645636e-2*2187^n/n^.500209\ 2648*(1.+.2880811429/n+.4096502691/n^2-3.408647005/n^3+15.69434840/n^4-2.764158\ 078/n^5), .8107222577e-2*2187^n/n^.5001922667*(1.+.2181694970/n+.3276023275/n^2 -3.200998892/n^3+14.98795469/n^4-5.072477048/n^5), .2431968981e-1*2187^n/n^.500\ 1765832*(1.+.1481444343/n+.2624302314/n^2-2.987484609/n^3+14.28397170/n^4-6.997\ 954367/n^5), .7295359528e-1*2187^n/n^.5001621195*(1.+.7801365530e-1/n+.21405091\ 02/n^2-2.773270655/n^3+13.58093355/n^4-8.587539904/n^5)] List must be at least of length 20 List must be at least of length 20 List must be at least of length 20 List must be at least of length 20 List must be at least of length 20 List must be at least of length 20 List must be at least of length 20 List must be at least of length 20 Theorem : Let a(n) be the number of ways of walking n steps, in the, 3, -dimensional Manhattan lattice with unit positive steps Always staying in the region 0 <= 3 x[1] - 3 x[2], 0 <= 3 x[2] - 2 x[3] The first, 150, terms of this sequence, starting at n=1 are [1, 2, 4, 9, 21, 51, 132, 349, 943, 2566, 7029, 19328, 53352, 148725, 417397, 1180247, 3352140, 9553969, 27283945, 78030323, 223808750, 643506549, 1855837600, 5364524552, 15538296691, 45069031580, 130856311772, 380461155633, 1107510640257, 3228658814870, 9424220983557, 27541616133186, 80563724012089, 235836891000373, 690955322275046, 2025854104169871, 5944753120211264, 17457910890822889, 51307085523061244, 150882356615197068, 443949228661424816, 1307001398238475097, 3849803088590742665, 11345935062797498804, 33455176877941067105, 98697388675778628368, 291301209074548527055, 860103822000637440820, 2540601811636867553891, 7507273568671633359497, 22191982460975888214902, 65624696284496033689322, 194130828266426036517000, 574465458087300212566418, 1700446137242221603081466, 5034928462008059671791849, 14912294219821326414739898, 44179411673683208895623603, 130922015942958816755566893, 388080198141883086157831353, 1150636218836256680117635429, 3412358506722930949056746919, 10122129331635556229108642631, 30031901279000411418206503845, 89122828290505281263695385283, 264536477137707908573090406274, 785364923252829663037514788644, 2332069309491916887968995501446, 6926127150279022518793421448693, 20573987891292856909387718439821, 61125287470259505076251576347219, 181634466018766230019498679675065, 539817858502746651648699219980974, 1604600284023987025322840764292866, 4770386197605810957729223610437147, 14184169750638238449203202142474960, 42181080717038385065189521435209582, 125456081497632359326938212015437397, 373186568283863572836335763197931147, 1110243455873242913450161727388134705, 3303448204794049113895745695032642288, 9830403599428371100191801495146208352, 29256839644206685636671627198050541676, 87083381815573227177805958834320105261, 259234454718266620207559746569485678628, 771790702517058905147355043387516027269, 2298021206548931399867209200297984447688, 6843140404072484875687259198245416968092, 20379897723657592041660163472841794786797, 60700459631135395932126054424876285010266, 180811048144179246851099370110094244304517, 538640880298502922014335993694965040491171, 1604777220360545759701013563791621754270445, 4781564281930619901722551388697216193147775, 14248345121676768461331904337870084002964503, 42461633977890970958286682056151953974838778, 126550995095660969478734973575300656656756758, 377199060673422812964363609444978373893290837, 1124373473222154164399033278028500659979569564, 3351855044749589081526612966090660825784143074, 9992945827388853564320869918822377502282872097, 29794422461415243073662347986707927424488343828, 88839997236018689484818642594378633353809797338, 264919098222301232181516318492568618747972113580, 790039469055625191609073453317767211184545319066, 2356210912492366318612894643752458650090428515856, 7027633302695909584513761756508169636814633948945, 20962004507649328308224130756474788981604278588994, 62529486826714600941029336660073877204042931919885, 186536780696144183393895719976123422852222837540718, 556507350056348629010797518091780252952796827894069, 1660365900859998719581862351489864807493644038855858, 4954073600565721400589526026151773467161121127559668, 14782455074497741598680435412402799010349069735038407, 44111877132049477628821596781942770024620668135696260, 131640336401966771198164402502379485847683135399796438, 392867668009706425186737200400276816886174528216811719, 1172537845029317246469070643925134443535798404929721825, 3499696797034170482131638978608752958196472531290356696, 10446152675513060526469006503509131466719970929074298076, 31182035048575270938144173010108549799564240618511587987, 93083818706760268940939156356800528972102563964172392750, 277885092431731187939884814692783443377008233045503312052, 829616037158535075340646561686390972447526931176904049179, 2476905233983235736191290268667901609334803363513285963569, 7395401161263263385128136514373538724334083775581519920699, 22081761667885981293134775397839736364294590131474790557628, 65936379715269419542459062092958695857522443710275401798464, 196895359366408518457465476406911173032280091679787180129526, 587982746472270772140373866424242584382210626905051693968878, 1755949447704652782823702396298528152121451324659145492064654, 5244177435997688828117247635798976337529063926615736892851212, 15662477222953620013078336661469630505565802418055375278244436, 46780069235086271757212838630015681631250747578030155374076057, 139726374194925571041688729994322101214650567915947161331603635, 417361816013621924218097757764910400364922922252466730515020741, 1246704677140900275072401189132990018853727876681982964851469448, 3724180235771034239833868787532754659295638441662864297655269672, 11125350603909374196013956797793645728020047044947540411807952506, 33236281329144333825997778176887678834452930016394430029629643533, 99294809002686307378901812177687687139513666662617460726055241018, 296657868934618814176248320050074043340897936575658302885652565587, 886339568438684739349066790372374017668255424938244918939512504928, 2648250908248738933611734530704587919498100668100951857074416007549, 7912843561925783377260573062631866941096086125247173251648299312503, 23643960569388981683093727243450071995168674072570042275465092202643, 70651581327990801606364588661847608152552701266031314145999213490791, 211123854196330878183079637165133860923305493865312302144222935859519, 630908407921365182750013360228430510337941053269850727503549355988394, 1885422504649687841218973560446851304244923118647562404208017465358512] Assuming that The connective constants for the number of steps with i mod, 8, for i from 1 to M are is, 6561, as it should The estimated critical expoents for each of these moduli are, [FAIL[1], FAIL[1], FAIL[1], FAIL[1], FAIL[1], FAIL[1], FAIL[1], FAIL[1]] The estimated asymptotics are [FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2]] Here is a table of the critical exponents [[[1, 1, 1], [-1.500000000, -1.500000000, -1.500000000]], [[2, 1, 1], [-.451297\ 4081, -.6025048236, -.5149470208, -.5031268982]], [[2, 2, 1], [-.4668690829, -.\ 5204045726, -.4737056073, -.5183056192, -.4759496325]], [[3, 1, 1], [-.49999691\ 33, -.4999498238, -.4998912795, -.4999240212, -.5000878345]], [[3, 2, 1], [-.\ 9800604700e-2, -.9678525463e-2, -.9551294235e-2, -.9441700854e-2, -.9335337218e\ -2, -.9230663113e-2]], [[3, 2, 2], [-.2324529943, -.5977241669, -.4951931489, -\ .5255723829, -.5017489070, -.4861944638, -.6976022861]], [[3, 3, 1], [-.5001368\ 669, -.5001269985, -.5001177011, -.5002092648, -.5001922667, -.5001765832, -.50\ 01621195]], [[3, 3, 2], [FAIL[1], FAIL[1], FAIL[1], FAIL[1], FAIL[1], FAIL[1], FAIL[1], FAIL[1]]]] ------------------------------- This took , 8507.485, seconds.