A193828 Even generalized pentagonal numbers.

0, 2, 12, 22, 26, 40, 70, 92, 100, 126, 176, 210, 222, 260, 330, 376, 392, 442, 532, 590, 610, 672, 782, 852, 876, 950, 1080, 1162, 1190, 1276, 1426, 1520, 1552, 1650, 1820, 1926, 1962, 2072, 2262, 2380, 2420, 2542, 2752, 2882, 2926, 3060, 3290, 3432, 3480

Offset

0,2

Comments

Even numbers in A001318.

Exponents in the expansion of Sum_{n >= 0} q^(2*n)/(Product_{k = 1..2*n} 1 + q^(2*k)) = 1 + q^2 - q^12 - q^22 + q^26 + q^40 - - + + ... (follows from Berndt et al., Theorem 3.3). Cf. A067589. - Peter Bala, Jan 21 2025

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.

Mircea Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157 (December 2015), Pages 223-232.

Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Formula

a(n) = A000217(A108752(n+1))/3 = 2*A154293(n+1).

G.f.: -2*x*(x^2-x+1)*(x^2+4*x+1)/((x-1)^3*(x^2+1)^2). - Colin Barker, Sep 12 2012

Sum_{n>=1} 1/a(n) = 6 - (1+4/sqrt(3))*Pi/2. - Amiram Eldar, Mar 18 2022

Mathematica

CoefficientList[Series[-2*x*(x^2 - x + 1)*(x^2 + 4*x + 1)/((x - 1)^3*(x^2 + 1)^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 06 2017 *)
LinearRecurrence[{3, -5, 7, -7, 5, -3, 1}, {0, 2, 12, 22, 26, 40, 70}, 50] (* Harvey P. Dale, Apr 09 2019 *)

Prog

(PARI) my(x='x+O('x^50)); concat([0], Vec(-2*x*(x^2-x+1)*(x^2+4*x+1)/((x-1)^3*(x^2+1)^2))) \\ G. C. Greubel, Jun 06 2017

Crossrefs

Cf. A001318, A067589, A154293.

Sequence in context: A077410 A211029 A225188 * A191226 A063599 A163479

Adjacent sequences: A193825 A193826 A193827 * A193829 A193830 A193831

Keyword

nonn,easy

Author

Omar E. Pol, Aug 19 2011

Status

approved