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

0,2

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

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

Formula

G.f.: (Sum_{k=-infinity..infinity} x^(k*(3*k-1)/2))^3.

G.f.: (Sum_{k>=0} x^A001318(k))^3.

Example

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].

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]

Crossrefs

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

Sequence in context: A256936 A021276 A329516 * A067753 A129023 A188883

Adjacent sequences: A290940 A290941 A290942 * A290944 A290945 A290946

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 14 2017

Status

approved