

A255916


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


2



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
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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 jgonal number and a kgonal 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 jgonal number and a kgonal 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

ZhiWei Sun, Table of n, a(n) for n = 0..10000


EXAMPLE

a(60) = 1 since 60 = (2)(5*(2)3)/2 + 1*(3*12) + 4*(7*45)/2.
a(279) = 1 since 279 = 3*(5*33)/2 + 0*(3*02) + 9*(7*95)/2.


MATHEMATICA

HQ[n_]:=HQ[n]=IntegerQ[Sqrt[40n+9]]&&(Mod[Sqrt[40n+9]+3, 10]==0Mod[Sqrt[40n+9]3, 10]==0)
Do[r=0; Do[If[HQ[nx(3x2)y(7y5)/2], r=r+1], {x, 0, (Sqrt[3n+1]+1)/3}, {y, 0, (Sqrt[56(nx(3x2))+25]+5)/14}];
Print[n, " ", r]; Continue, {n, 0, 100}]


CROSSREFS

Cf. A000567, A001106, A085787.
Sequence in context: A230206 A285117 A135910 * A107333 A161642 A152141
Adjacent sequences: A255913 A255914 A255915 * A255917 A255918 A255919


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 11 2015


STATUS

approved



