Here is an article, regarding the mod 3 analog of A246039 and A246038, type\ : 2 2 2 2 2 n On the sequence, (x y + x y + x + x y + y + y + 1) , modulo , 3, evaluated at , {x = 1, y = 1} By Shalosh B. Ekhad The first, 41, terms staring at n=0 are [1, 7, 31, 7, 49, 121, 31, 217, 307, 7, 49, 217, 49, 343, 679, 121, 847, 1057, 31, 217, 529, 217, 1519, 1639, 307, 2149, 2917, 7, 49, 217, 49, 343, 847, 217, 1519, 2149, 49, 343, 1519, 343, 2401] Just for kicks, the googol-th term of our sequence is 6306968689907234090754806322547611881699018447996602403254197248335480197855823\ 452831932958171844417370314407708021484375 i The first , 40, terms of the sparse subsequence at the, 3 - 1, places are [1, 31, 307, 2917, 26557, 240379, 2166079, 19506937, 175576897, 1580283991, 14222499811, 128002760341, 1152020187013, 10368168492307, 93313380132775, 839819901950929, 7558375679520073, 68025366405346255, 612228214084302571, 5510053546445685901, 49590479902037385997, 446314309617808027531, 4016828737901704557103, 36151458407219547138121, 325363124488193675549905, 2928268114673557609882183, 26354413003552671942181747, 237189716892525218728868101, 2134707451341219085599962773, 19212367058676810187006535683, 172911303511305947325055235767, 1556201731519206908736120285889, 14005815583265242779528715547737, 126052340247380481513430320659647, 1134471062216523093631197971607547, 10210239559899935855344504892471677, 91892156038858883096531216180484637, 827029404348544708599468964456250491, 7443264639131058267390074598968161087, 66989381752150722993061660162522939705, 602904435769214513104334780514484333537 ] Using the found enumerative automaton with, 34, states, that we omit, it follows that the (rigorously) PROVED rational generating function for that sparse subsequ\ ence is 7 6 5 4 3 2 3564 t - 3132 t + 1296 t - 384 t + 131 t + 91 t - 17 t - 1 - --------------------------------------------------------------------- 3 2 2 (t + 1) (3 t - 1) (9 t - 1) (18 t - 4 t + 5 t - 1) (6 t - 2 t - 1) and in Maple notation -(3564*t^7-3132*t^6+1296*t^5-384*t^4+131*t^3+91*t^2-17*t-1)/(t+1)/(3*t-1)/(9*t-\ 1)/(18*t^3-4*t^2+5*t-1)/(6*t^2-2*t-1) This ends this article, that took, 0.109, seconds. Here is an article, regarding mod 3 analog of A253069 and A253070 2 2 2 n On the sequence, (x y + x + x y + y + y + 1) , modulo , 3, evaluated at , {x = 1, y = 1} By Shalosh B. Ekhad The first, 41, terms staring at n=0 are [1, 6, 27, 6, 36, 111, 27, 162, 234, 6, 36, 162, 36, 216, 597, 111, 666, 903, 27, 162, 366, 162, 972, 1389, 234, 1404, 2196, 6, 36, 162, 36, 216, 666, 162, 972, 1404, 36, 216, 972, 216, 1296] Just for kicks, the googol-th term of our sequence is 1509260199812017038627969881957065714111146448793996776307350627340877606172308\ 32747084666696457247892231570262589440 i The first , 40, terms of the sparse subsequence at the, 3 - 1, places are [1, 27, 234, 2196, 19470, 171774, 1502274, 13132101, 114612924, 1000250709, 8726918490, 76139244333, 664252349808, 5795043834816, 50556392779749, 441057531210867, 3847809926066685, 33568500532532607, 292853302608046581, 2554867036207936197, 22288787941221205296, 194448501369278091558, 1696378436496075040866, 14799290187048759027255, 129109746296456861868219, 1126359871071599114522787, 9826419732292707780243972, 85726176183425047429041546, 747879439580085421850728305, 6524537557210734930440162766, 56920391284911007833655631085, 496576334432300598528374246613, 4332156725403927981762327520809, 37793951487672826912610491003053, 329716319050912150821715349538471, 2876461623335151009217732457855439, 25094394764371621737148406536307448, 218924752369883324645620974407361867, 1909910466071924202052713380047083258, 16662154228444761552360111487532921373, 145361465086617142843689733818474176340 ] Using the found enumerative automaton with, 66, states, that we omit, it follows that the (rigorously) PROVED rational generating function for that sparse subsequ\ ence is 64 63 62 61 - (74281925718 t + 42762524490 t - 347510823759 t - 112903693812 t 60 59 58 - 175706111280 t + 891798104567 t + 1947149644151 t 57 56 55 - 8371954755965 t - 24117337030508 t + 48298578731565 t 54 53 52 + 38277916863088 t - 68204657649324 t - 24424288817290 t 51 50 49 - 17576777015233 t + 52892016096006 t + 123662583718887 t 48 47 46 - 108688055419701 t - 134383756993949 t + 121805289289270 t 45 44 43 + 45479223115244 t - 69650606500019 t + 56420216882294 t 42 41 40 - 18693008429726 t - 80314924219773 t + 75017099454042 t 39 38 37 + 31226000704372 t - 63058240809561 t + 13517628651982 t 36 35 34 + 21260910127026 t - 17942853527722 t + 2443914771274 t 33 32 31 + 5445540516344 t - 3953469078612 t + 842275407375 t 30 29 28 27 + 394077756719 t - 718552159244 t + 529189259031 t - 61839295683 t 26 25 24 23 - 174507997458 t + 120856176198 t - 5506525011 t - 31002299946 t 22 21 20 19 + 9210529982 t + 3505581436 t - 182642952 t - 570940679 t 18 17 16 15 - 654861817 t + 154753570 t + 234614664 t - 18241055 t 14 13 12 11 10 - 54760049 t - 2059430 t + 9612138 t + 1287663 t - 936904 t 9 8 7 6 5 4 3 - 636617 t + 155612 t + 130463 t - 40162 t - 8944 t + 4286 t + 16 t 2 / 32 31 30 - 158 t + 11 t + 1) / ((776160 t + 2563323 t - 38469 t / 29 28 27 26 25 - 1850036 t - 22941902 t + 12755157 t + 20927273 t - 18104351 t 24 23 22 21 20 - 4657554 t + 2830490 t - 2182772 t + 3108349 t + 4250259 t 19 18 17 16 15 - 3000475 t - 1947760 t + 3052354 t - 2629980 t + 275317 t 14 13 12 11 10 + 1700236 t - 1157376 t + 37932 t + 278782 t - 163228 t 9 8 7 6 5 4 3 + 44622 t - 3363 t - 5512 t + 7960 t - 5110 t + 1318 t + 54 t 2 33 32 31 30 - 98 t + 18 t - 1) (32076 t - 70890 t - 152337 t + 89911 t 29 28 27 26 25 - 565296 t + 682346 t - 348143 t - 924863 t + 1295637 t 24 23 22 21 20 - 382656 t + 1015236 t + 88498 t - 2851511 t + 669541 t 19 18 17 16 15 + 2351261 t - 558618 t - 1111198 t + 5430 t + 344843 t 14 13 12 11 10 9 + 225900 t - 83388 t - 127014 t - 348 t + 37064 t + 8692 t 8 7 6 5 4 3 2 - 5523 t - 3368 t + 98 t + 600 t + 134 t - 54 t - 22 t + 2 t + 1)) and in Maple notation -(74281925718*t^64+42762524490*t^63-347510823759*t^62-112903693812*t^61-\ 175706111280*t^60+891798104567*t^59+1947149644151*t^58-8371954755965*t^57-\ 24117337030508*t^56+48298578731565*t^55+38277916863088*t^54-68204657649324*t^53 -24424288817290*t^52-17576777015233*t^51+52892016096006*t^50+123662583718887*t^ 49-108688055419701*t^48-134383756993949*t^47+121805289289270*t^46+ 45479223115244*t^45-69650606500019*t^44+56420216882294*t^43-18693008429726*t^42 -80314924219773*t^41+75017099454042*t^40+31226000704372*t^39-63058240809561*t^ 38+13517628651982*t^37+21260910127026*t^36-17942853527722*t^35+2443914771274*t^ 34+5445540516344*t^33-3953469078612*t^32+842275407375*t^31+394077756719*t^30-\ 718552159244*t^29+529189259031*t^28-61839295683*t^27-174507997458*t^26+ 120856176198*t^25-5506525011*t^24-31002299946*t^23+9210529982*t^22+3505581436*t ^21-182642952*t^20-570940679*t^19-654861817*t^18+154753570*t^17+234614664*t^16-\ 18241055*t^15-54760049*t^14-2059430*t^13+9612138*t^12+1287663*t^11-936904*t^10-\ 636617*t^9+155612*t^8+130463*t^7-40162*t^6-8944*t^5+4286*t^4+16*t^3-158*t^2+11* t+1)/(776160*t^32+2563323*t^31-38469*t^30-1850036*t^29-22941902*t^28+12755157*t ^27+20927273*t^26-18104351*t^25-4657554*t^24+2830490*t^23-2182772*t^22+3108349* t^21+4250259*t^20-3000475*t^19-1947760*t^18+3052354*t^17-2629980*t^16+275317*t^ 15+1700236*t^14-1157376*t^13+37932*t^12+278782*t^11-163228*t^10+44622*t^9-3363* t^8-5512*t^7+7960*t^6-5110*t^5+1318*t^4+54*t^3-98*t^2+18*t-1)/(32076*t^33-70890 *t^32-152337*t^31+89911*t^30-565296*t^29+682346*t^28-348143*t^27-924863*t^26+ 1295637*t^25-382656*t^24+1015236*t^23+88498*t^22-2851511*t^21+669541*t^20+ 2351261*t^19-558618*t^18-1111198*t^17+5430*t^16+344843*t^15+225900*t^14-83388*t ^13-127014*t^12-348*t^11+37064*t^10+8692*t^9-5523*t^8-3368*t^7+98*t^6+600*t^5+ 134*t^4-54*t^3-22*t^2+2*t+1) This ends this article, that took, 1.746, seconds.