A255916 Number of ways to write n as the sum of a generalized heptagonal number, an octagonal number and a nonagonal number.

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

Offset: 0

Author

Zhi-Wei Sun, Mar 11 2015

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n. Moreover, for k >= j >=3, every nonnegative integer can be written as the sum of a generalized heptagonal number, a j-gonal number and a k-gonal number, if and only if (j,k) is among the following ordered pairs:

(3,k) (k = 3..19, 21..24, 26, 27, 29, 30), (4,k) (k = 4..11, 13, 14, 17, 19, 20, 23, 26), (5,6), (5,9), (6,7), (8,9).

(ii) For k >= j >= 3, every nonnegative integer can be written as the sum of a generalized pentagonal number, a j-gonal number and a k-gonal number, if and only if (j,k) is among the following ordered pairs:

(3,k) (k = 3..20, 22, 24, 25, 28..30, 32, 37), (4,k) (k = 4..13, 15, 16, 18, 20..25, 27, 28, 31, 33, 34), (5,k) (k = 6..12, 20), (6,k) (k = 7..10), (7,9), (7,11), (8,10), (9,11).

Examples

			 a(60) = 1 since 60 = (-2)(5*(-2)-3)/2 + 1*(3*1-2) + 4*(7*4-5)/2.
a(279) = 1 since 279 = 3*(5*3-3)/2 + 0*(3*0-2) + 9*(7*9-5)/2.
		

Links

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

Crossrefs

Cf. A000567, A001106, A085787.

Programs

Mathematica

HQ[n_]:=HQ[n]=IntegerQ[Sqrt[40n+9]]&&(Mod[Sqrt[40n+9]+3,10]==0||Mod[Sqrt[40n+9]-3,10]==0)
Do[r=0;Do[If[HQ[n-x(3x-2)-y(7y-5)/2],r=r+1],{x,0,(Sqrt[3n+1]+1)/3},{y,0,(Sqrt[56(n-x(3x-2))+25]+5)/14}];
Print[n," ",r];Continue,{n,0,100}]