A255934 -id:A255934 - cp's OEIS frontend

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

Zhi-Wei Sun, Mar 17 2015

nonn

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.

			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.

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.

Cf. A085787, A255934.

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}]

A290943 Number of ways to write n as an ordered sum of 3 generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 3, 6, 7, 6, 6, 7, 12, 12, 12, 9, 6, 12, 12, 18, 13, 12, 18, 12, 18, 12, 13, 18, 12, 24, 12, 12, 24, 21, 30, 12, 18, 18, 12, 24, 18, 19, 18, 24, 24, 18, 24, 36, 24, 18, 19, 18, 24, 24, 30, 18, 12, 36, 30, 24, 21, 18, 36, 24, 36, 24, 12, 36, 36, 36, 18, 25, 30, 24, 24, 24, 30, 24, 36, 30, 24

Offset: 0

Ilya Gutkovskiy, Aug 14 2017

Conjecture: every number is the sum of at most k - 4 generalized k-gonal numbers (for k >= 8).

In 1830, Legendre showed that for each integer m>4 every integer N >= 28*(m-2)^3 can be written as the sum of five m-gonal numbers. In 1994 R. K. Guy proved that each natural number is the sum of three generalized pentagonal numbers. In a 2016 paper Zhi-Wei Sun proved that each natural number is the sum of four octagonal numbers. - Zhi-Wei Sun, Oct 03 2020

			a(6) = 7 because we have [5, 1, 0], [5, 0, 1], [2, 2, 2], [1, 5, 0], [1, 0, 5], [0, 5, 1] and [0, 1, 5].

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers

Cf. A001318, A002175, A008443, A080995, A093518, A093519, A255350, A255934, A256132, A256171, A280718.

N:= 100;
bds:= [fsolve(k*(3*k-1)/2 = N)];
G:= add(x^(k*(3*k-1)/2),k=floor(min(bds))..ceil(max(bds)))^3:
seq(coeff(G,x,n),n=0..N); # Robert Israel, Aug 16 2017

nmax = 75; CoefficientList[Series[Sum[x^(k (3 k - 1)/2), {k, -nmax, nmax}]^3, {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[Sum[x^((6 k^2 + 6 k + (-1)^(k + 1) (2 k + 1) + 1)/16), {k, 0, nmax}]^3, {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[EllipticTheta[4, 0, x^3]^3/QPochhammer[x, x^2]^3, {x, 0, nmax}], x]

G.f.: (Sum_{k=-infinity..infinity} x^(k*(3*k-1)/2))^3.
G.f.: (Sum_{k>=0} x^A001318(k))^3.
G.f.: Product_{n >= 1} ( (1 - q^(3*n))/(1 - q^n + q^(2*n)) )^3. - Peter Bala, Jan 04 2025

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A256171 Number of ways to write n as the sum of three unordered generalized heptagonal numbers.

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