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

0,2

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).

Links

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

Example

 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.

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}]

Crossrefs

Cf. A000567, A001106, A085787.

Sequence in context: A230206 A285117 A135910 * A367825 A107333 A161642

Adjacent sequences: A255913 A255914 A255915 * A255917 A255918 A255919

Keyword

nonn

Author

Zhi-Wei Sun, Mar 11 2015

Status

approved