#!/usr/local/bin/maple # -*- maplev -*- # Nathaniel Shar # HW 7 # Experimental Mathematics # It is okay to link to this assignment on the course webpage. Help := proc(): print(`Gsomosk(Ini, k, n), GenHomPol(vars, d, a), GenNonHomPol(vars, d, a)`): ############# # Problem 1 # ############# Gsomosk := proc(Ini, k, n) option remember: if n <= 0 then: return FAIL: elif n <= k then: return Ini[n]: else: return add(Gsomosk(Ini, k, n-i)*Gsomosk(Ini, k, n-k+i), i=1..floor(k/2))/Gsomosk(Ini, k, n-k): fi: end: ############# # Problem 2 # ############# # Running # seq(hastype(Gsomosk([1\$i], i, 2*i+5), integer), i=4..100); # yields a list starting with 4 "true"s, followed by a bunch of # "false"s. Generating the Somos k-sequences for k from 4 through 7 # yields sequences that appear to be all integers, so the experimental # evidence suggests that the Somos k-sequence consists of all integers # iff 4 <= k <= 7. ############# # Problem 3 # ############# # I was not able to gather an adequate amount of data as maple # repeatedly hung while attempting the calculations. However, I was able to # discover that for 4 <= k <= 7, the sequence appears to consist # entirely of Laurent polynomials, while for k = 8 and k = 9, some # terms of the sequence are not Laurent polynomials. I would # conjecture that the sequence eventually contains a rational function # that is not a Laurent polynomial for k >= 8. ############# # Problem 4 # ############# GenHomPol := proc(vars,d,a, start := 1) local m, firstvar, newstart, i: m := nops(vars): if m <= 1 then: return a[start]*vars[1]^d: fi: firstvar := vars[1]: newstart := k -> add((i+m-2)!/i!/(m-2)!, i=0..k-1): return add(firstvar^(d-i)*GenHomPol(i, vars[2..m], a, start+newstart(i)), i=0..d): end: ############# # Problem 5 # ############# GenNonHomPol := proc(vars,d, a) local m, start: m := nops(vars): start := k -> add(binomial(i+m-1, i), i=0..k-1)+1: return add(GenHomPol(i, vars, a, start(i)), i=0..d): end: