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A255916 Number of ways to write n as the sum of a generalized heptagonal number, an octagonal number and a nonagonal number. 2

%I

%S 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,

%T 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,

%U 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

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

%C 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:

%C (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).

%C (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:

%C (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).

%H Zhi-Wei Sun, <a href="/A255916/b255916.txt">Table of n, a(n) for n = 0..10000</a>

%e a(60) = 1 since 60 = (-2)(5*(-2)-3)/2 + 1*(3*1-2) + 4*(7*4-5)/2.

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

%t HQ[n_]:=HQ[n]=IntegerQ[Sqrt[40n+9]]&&(Mod[Sqrt[40n+9]+3,10]==0||Mod[Sqrt[40n+9]-3,10]==0)

%t 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}];

%t Print[n," ",r];Continue,{n,0,100}]

%Y Cf. A000567, A001106, A085787.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Mar 11 2015

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