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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

Ilya Gutkovskiy, Extended graphical example

Eric Weisstein's World of Mathematics, Pentagonal Number

Index to sequences related to polygonal numbers

FORMULA

G.f.: (Sum_{k=-inf..inf} 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: A016616 A256936 A021276 * A067753 A129023 A188883

Adjacent sequences:  A290940 A290941 A290942 * A290944 A290945 A290946

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Aug 14 2017

STATUS

approved

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Last modified February 23 21:23 EST 2018. Contains 299588 sequences. (Running on oeis4.)