A280719 Expansion of (Sum_{k>=0} x^(k*(2*k-1)))^6.

1, 6, 15, 20, 15, 6, 7, 30, 60, 60, 30, 6, 15, 60, 90, 66, 45, 60, 80, 90, 66, 50, 120, 180, 135, 60, 15, 60, 186, 210, 141, 126, 120, 126, 165, 180, 241, 300, 210, 90, 90, 180, 270, 270, 210, 212, 270, 270, 200, 210, 366, 450, 390, 270, 135, 210, 375, 360, 396, 420, 300, 330, 375, 380, 510, 480, 336, 450, 510, 390, 330

Offset: 0

Author

Ilya Gutkovskiy, Feb 10 2017

Keywords

• nonn

Comments

Number of ways to write n as an ordered sum of 6 hexagonal numbers (A000384).

a(n) > 0 for all n >= 0.

Every number is the sum of at most 6 hexagonal numbers.

Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

Examples

			a(6) = 7 because we have:
[6, 0, 0, 0, 0, 0]
[0, 6, 0, 0, 0, 0]
[0, 0, 6, 0, 0, 0]
[0, 0, 0, 6, 0, 0]
[0, 0, 0, 0, 6, 0]
[0, 0, 0, 0, 0, 6]
[1, 1, 1, 1, 1, 1]
		

Links

• Ilya Gutkovskiy, Extended graphical example
• Eric Weisstein's World of Mathematics, Hexagonal Number
• Index to sequences related to polygonal numbers

Crossrefs

Cf. A000384, A007536, A008440, A045848, A280718, A282248.

Programs

Mathematica

nmax = 70; CoefficientList[Series[Sum[x^(k (2 k - 1)), {k, 0, nmax}]^6, {x, 0, nmax}], x]

Formula

G.f.: (Sum_{k>=0} x^(k*(2*k-1)))^6.