OFFSET
1,2
COMMENTS
Conjecture: a(n) > 0 for all n > 0. In other words, each positive integer n can be written as the sum of a positive generalized pentagonal number, a tetrahedral number and a tetrahedral number times three.
This has been verified for all n = 1..2*10^7.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(59) = 1 with 59 = (-3)*(3*(-3)+1)/2 + 2*3*4/2 + 5*6*7/6.
a(19694) = 1 with 19694 = 20*(3*20+1)/2 + 10*11*12/2 + 47*48*49/6.
a(19919) = 1 with 19919 = (-45)*(3*(-45)+1)/2 + 30*31*32/2 + 22*23*24/6.
a(33989) = 1 with 33989 = 55*(3*55+1)/2 + 20*21*22/2 + 52*53*54/6.
a(60769) = 1 with 60769 = 46*(3*46+1)/2 + 47*48*49/2 + 23*24*25/6.
MATHEMATICA
f[n_]:=f[n]=Binomial[n+2, 3]; PQ[n_]:=PQ[n]=IntegerQ[Sqrt[24n+1]];
tab={}; Do[r=0; Do[If[f[x]>=n/3, Goto[cc]]; Do[If[f[y]>=n-3*f[x], Goto[bb]]; If[PQ[n-3*f[x]-f[y]], r=r+1]; Label[aa], {y, 0, n-1-3*f[x]}]; Label[bb], {x, 0, (n-1)/3}]; Label[cc]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 10 2019
STATUS
approved