A266677 Alternating sum of hexagonal pyramidal numbers.

0, -1, 6, -16, 34, -61, 100, -152, 220, -305, 410, -536, 686, -861, 1064, -1296, 1560, -1857, 2190, -2560, 2970, -3421, 3916, -4456, 5044, -5681, 6370, -7112, 7910, -8765, 9680, -10656, 11696, -12801, 13974, -15216, 16530, -17917, 19380, -20920, 22540

Offset

0,3

Comments

More generally, the ordinary generating function for the alternating sum of k-gonal pyramidal numbers is x*(1 + (3 - k)*x)/((x - 1)*(x + 1)^4).

Links

Table of n, a(n) for n=0..40.

Ilya Gutkovskiy, Extended graphic representation

OEIS Wiki, Figurate numbers

Eric Weisstein's World of Mathematics, Pyramidal Number

Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number

Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Formula

G.f.: x*(1 - 3*x)/((x - 1)*(x + 1)^4).

a(n) = ((-1)^n*(2*n*(n + 2)*(4*n + 1) - 3) + 3)/24.

a(n) = Sum_{k = 0..n} (-1)^k*A002412(k).

Mathematica

Table[((-1)^n (2 n (n + 2) (4 n + 1) - 3) + 3)/24, {n, 0, 40}]
LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 6, -16, 34}, 40]

Prog

(PARI) concat(0, Vec(x*(1 - 3*x)/((x - 1)*(x + 1)^4) + O(x^50))) \\ Michel Marcus, Feb 02 2016

Crossrefs

Cf. A000292, A002412, A002717, A173196.

Sequence in context: A361230 A236773 A131820 * A083053 A083046 A192749

Adjacent sequences: A266674 A266675 A266676 * A266678 A266679 A266680

Keyword

sign,easy

Author

Ilya Gutkovskiy, Feb 02 2016

Status

approved