

A290943


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


1



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
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OFFSET

0,2


COMMENTS

Conjecture: every number is the sum of at most k  4 generalized kgonal numbers (for k >= 8).
In 1830, Legendre showed that for each integer m>4 every integer N >= 28*(m2)^3 can be written as the sum of five mgonal numbers. In 1994 R. K. Guy proved that each natural number is the sum of three generalized pentagonal numbers. In a 2016 paper ZhiWei Sun proved that each natural number is the sum of four octagonal numbers.  ZhiWei Sun, Oct 03, 2020


LINKS

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
ZhiWei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190211.
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*k1)/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*k1)/2 = N)];
G:= add(x^(k*(3*k1)/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



