A307981 Number of ways to write n as x^3 + 2*y^3 + 3*z^3 + w*(w+1)*(w+2)/6, where x,y,z,w are nonnegative integers.

1, 2, 2, 3, 4, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 2, 1, 5, 4, 1, 4, 4, 4, 4, 5, 6, 3, 5, 5, 2, 4, 4, 3, 5, 5, 3, 3, 4, 3, 3, 2, 5, 3, 3, 5, 2, 2, 3, 3, 5, 2, 4, 4, 3, 3, 5, 6, 3, 5, 6, 3, 4, 4, 5, 7, 5, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 3, 4, 5, 6, 8, 7, 7, 6, 7, 8, 6, 7, 4, 5, 4, 4, 2, 2, 4, 4, 5, 4

Offset: 0

Author

Zhi-Wei Sun, May 08 2019

Keywords

• nonn
• look

Comments

Conjecture: a(n) > 0 for every nonnegative integer n. In other words, we have {x^3 + 2*y^3 + 3*z^3 + w*(w+1)*(w+2)/6: x,y,z,w = 0,1,2,...} = {0,1,2,...}.

We have verified a(n) > 0 for all n = 0..2*10^6.

Examples

			a(19) = 1 with 19 = 0^3 + 2*2^3 + 3*1^3 + 0*1*2/6.
a(22) = 1 with 22 = 0^3 + 2*1^3 + 3*0^3 + 4*5*6/6.
a(112) = 1 with 112 = 3^3 + 2*0^3 + 3*3^3 + 2*3*4/6.
a(158) = 1 with 158 = 3^3 + 2*4^3 + 3*1^3 + 0*1*2/6.
a(791) = 1 with 791 = 1^3 + 2*5^3 + 3*5^3 + 9*10*11/6.
a(956) = 1 with 956 = 9^3 + 2*0^3 + 3*4^3 + 5*6*7/6.
a(6363) = 1 with 6363 = 10^3 + 2*13^3 + 3*0^3 + 17*18*19/6.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000292, A000578, A262813, A306460, A306477, A306790.

Programs

Mathematica

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];f[w_]:=f[w]=Binomial[w+2,3];
tab={};Do[r=0;w=0;Label[bb];If[f[w]>n,Goto[aa]];Do[If[CQ[n-f[w]-2y^3-3z^3],r=r+1],{y,0,((n-f[w])/2)^(1/3)},{z,0,((n-f[w]-2y^3)/3)^(1/3)}];w=w+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,0,100}];Print[tab]