OFFSET
0,3
COMMENTS
Conjecture: a(n) > 0 for all n. In other words, any nonnegative integer n can be expressed as the sum of three pentagonal numbers and a second pentagonal number.
See also A255350 for a similar conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 0..6000
Zhi-Wei Sun, A result similar to Lagrange's theorem, arXiv:1503.03743 [math.NT], 2015.
EXAMPLE
a(4) = 1 since 4 = 1*(3*1+1)/2 + 0*(3*0-1)/2 + 1*(3*1-1)/2 + 1*(3*1-1)/2.
a(11) = 1 since 11 = 0*(3*0+1)/2 + 1*(3*1-1)/2 + 2*(3*2-1)/2 + 2*(3*2-1)/2.
a(23) = 1 since 23 = 0*(3*0+1)/2 + 0*(3*0-1)/2 + 1*(3*1-1)/2 + 4*(3*4-1)/2.
MATHEMATICA
GenPen[n_]:=IntegerQ[Sqrt[24n+1]]&&Mod[Sqrt[24n+1], 6]==1
Do[r=0; Do[If[GenPen[n-x(3x-1)/2-y(3y-1)/2-z(3z-1)/2], r=r+1], {x, 0, (Sqrt[8n+1]+1)/6}, {y, x, (Sqrt[12(n-x(3x-1)/2)+1]+1)/6},
{z, y, (Sqrt[24(n-x(3x-1)/2-y(3y-1)/2)+1]+1)/6}]; Print[n, " ", r]; Continue, {n, 0, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 15 2015
STATUS
approved