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

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

Author

Ilya Gutkovskiy, Aug 14 2017

Keywords

• nonn
• easy

Comments

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

Examples

			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].
		

Links

• 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

Crossrefs

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

Programs

Maple

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

Mathematica

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]

Formula

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