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

0,9

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.

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.

Example

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.

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}]

Crossrefs

Cf. A085787, A255934.

Sequence in context: A287455 A216789 A097951 * A058643 A343748 A029368

Adjacent sequences: A256168 A256169 A256170 * A256172 A256173 A256174

Keyword

nonn

Author

Zhi-Wei Sun, Mar 17 2015

Status

approved