A306790 Number of ways to write n as x*(3*x+1)/2 + y*(y+1)*(y+2)/2 + z*(z+1)*(z+2)/6, where x is a nonzero integer, and y and z are nonnegative integers.

1, 2, 1, 1, 4, 3, 1, 3, 3, 1, 3, 3, 2, 4, 5, 3, 3, 4, 4, 2, 2, 4, 3, 3, 6, 3, 4, 4, 3, 3, 2, 4, 3, 3, 7, 5, 4, 5, 7, 3, 3, 6, 3, 4, 5, 3, 5, 4, 3, 3, 3, 6, 4, 3, 5, 3, 6, 5, 1, 5, 7, 6, 4, 5, 4, 6, 5, 2, 5, 7, 7, 5, 7, 5, 5, 3, 5, 3, 2, 6, 5, 7, 5, 1, 6, 5, 7, 3, 5, 6, 6, 6, 4, 3, 4, 7, 4, 3, 6, 6

Offset: 1

Author

Zhi-Wei Sun, Mar 10 2019

Keywords

• nonn

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.

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000292, A001318, A306460, A306477.

Programs

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]