OFFSET
0,2
COMMENTS
Number of ways to write n as an ordered sum of 6 square pyramidal numbers (A000330).
Conjecture: a(n) > 0 for all n.
Extended conjecture: every number is the sum of at most 6 square pyramidal numbers.
Generalized conjecture: every number is the sum of at most k+2 k-gonal pyramidal numbers (except k = 5). - Ilya Gutkovskiy, Feb 10 2017
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Square Pyramidal Number
FORMULA
G.f.: (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.
EXAMPLE
a(5) = 12 because we have:
[5, 0, 0, 0, 0, 0]
[0, 5, 0, 0, 0, 0]
[0, 0, 5, 0, 0, 0]
[0, 0, 0, 5, 0, 0]
[0, 0, 0, 0, 5, 0]
[0, 0, 0, 0, 0, 5]
[1, 1, 1, 1, 1, 0]
[1, 1, 1, 1, 0, 1]
[1, 1, 1, 0, 1, 1]
[1, 1, 0, 1, 1, 1]
[1, 0, 1, 1, 1, 1]
[0, 1, 1, 1, 1, 1]
MATHEMATICA
nmax = 69; CoefficientList[Series[(Sum[x^(k (k + 1) (2 k + 1)/6), {k, 0, nmax}])^6, {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 07 2017
STATUS
approved