A014633 Even pentagonal numbers.

0, 12, 22, 70, 92, 176, 210, 330, 376, 532, 590, 782, 852, 1080, 1162, 1426, 1520, 1820, 1926, 2262, 2380, 2752, 2882, 3290, 3432, 3876, 4030, 4510, 4676, 5192, 5370, 5922, 6112, 6700, 6902, 7526, 7740, 8400, 8626, 9322

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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

Formula

G.f.: 2*(6+5*x+12*x^2+x^3)/((1+x)^2*(1-x)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009, corrected by R. J. Mathar, Sep 16 2009

From Ant King, Aug 16 2011: (Start)

a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).

a(n) = 48+2*a(n-2)-a(n-4).

a(n) = 1/8*(1-3*(-1)^(n+1)+12*(n+1))*(1-(-1)^(n+1)+4*(n+1)).(End)

Sum_{n>=1} 1/a(n) = 3*log(3)/2 - (1/sqrt(3)+1/4)*Pi - sqrt(3)*log(2-sqrt(3))/2. - Amiram Eldar, Jan 13 2024

Mathematica

LinearRecurrence[{1, 2, -2, -1, 1}, {0, 12, 22, 70, 92}, 40] (* Harvey P. Dale, Aug 26 2014 *)
Select[PolygonalNumber[5, Range[0, 100]], EvenQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 15 2017 *)

Prog

(Magma) [1/8*(1-3*(-1)^(n+1)+12*(n+1))*(1-(-1)^(n+1)+4*(n+1)): n in [0..40]]; // Vincenzo Librandi, Aug 17 2011
(PARI) lista(nn) = {forstep (n=0, nn, 2, if (ispolygonal(n, 5), print1(n, ", ")); ); } \\ Michel Marcus, Jun 20 2015

Crossrefs

Cf. A000326, A014632.

Sequence in context: A200197 A115709 A115703 * A298323 A299216 A227072

Adjacent sequences: A014630 A014631 A014632 * A014634 A014635 A014636

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

More terms from Patrick De Geest

Status

approved