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

Author

Omar E. Pol, Aug 19 2011

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001318, A067589, A154293.

Programs

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 *)

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

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