A256171 Number of ways to write n as the sum of three unordered generalized heptagonal numbers.

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 0, 1, 2, 1, 2, 3, 0, 1, 3, 1, 2, 3, 1, 1, 1, 1, 3, 3, 1, 2, 1, 2, 3, 1, 2, 4, 2, 1, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 3, 2, 1, 2, 2, 3, 5, 2, 2, 2, 2, 3, 4, 2, 2, 4, 1, 3, 2, 1, 4, 3, 2, 2, 5, 2, 4, 3, 0, 4, 2, 1, 3, 6, 3, 3, 3, 1, 5, 2, 3, 5, 2, 2, 3, 3, 1, 5, 3, 1, 3, 3, 4

Offset: 0

Author

Zhi-Wei Sun, Mar 17 2015

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 except for n = 10, 16, 76, 307.

(ii) For any integer m > 2 not divisible by 4, each sufficiently large integer n can be written as the sum of three generalized m-gonal numbers.

In 1994 R. K. Guy noted that none of 10, 16 and 76 can be written as the sum of three generalized heptagonal numbers.

Examples

			a(157) = 1 since 157 = 3*(5*3-3)/2 + (-3)*(5*(-3)-3)/2 + 7*(5*7-3)/2.
a(748) = 1 since 748 = 0*(5*0-3)/2 + 0*(5*0-3)/2 + (-17)*(5*(-17)-3)/2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.
• Zhi-Wei Sun, A result similar to Lagrange's theorem, arXiv:1503.03743 [math.NT], 2015.

Crossrefs

Cf. A085787, A255934.

Programs

Mathematica

T[n_]:=Union[Table[x(5x-3)/2, {x, -Floor[(Sqrt[40n+9]-3)/10], Floor[(Sqrt[40n+9]+3)/10]}]]
L[n_]:=Length[T[n]]
Do[r=0;Do[If[Part[T[n],x]>n/3,Goto[aa]];Do[If[Part[T[n],x]+2*Part[T[n],y]>n,Goto[bb]];
If[MemberQ[T[n], n-Part[T[n],x]-Part[T[n],y]]==True,r=r+1];
Continue,{y,x,L[n]}];Label[bb];Continue,{x,1,L[n]}];Label[aa];Print[n," ",r];Continue, {n,0,100}]