A015223 Odd pentagonal pyramidal numbers.

1, 75, 405, 1183, 2601, 4851, 8125, 12615, 18513, 26011, 35301, 46575, 60025, 75843, 94221, 115351, 139425, 166635, 197173, 231231, 269001, 310675, 356445, 406503, 461041, 520251, 584325, 653455, 727833, 807651, 893101, 984375

Offset

0,2

Comments

Also first bisection of A139757. - Bruno Berselli, Feb 13 2012

Links

Bruno Berselli, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

G.f.: (1 + 71*x + 111*x^2 + 9*x^3)/(1-x)^4. - Colin Barker, Feb 13 2012

a(n) = (2n+1)*(4n+1)^2 = A130656(4n+1). - Bruno Berselli, Feb 13 2012

From Ant King, Oct 23 2012: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 192.

Sum_{n>=0} 1/a(n) = (8*C - 2*Pi + Pi^2 - 4*log(2))/8, where C is Catalan's constant (A006752). (End)

E.g.f.: (1 + 74*x + 128*x^2 + 32*x^3)*exp(x). - G. C. Greubel, Nov 04 2017

Mathematica

Table[((n+1)^3+(n+1)^2)/2, {n, 0, 200, 4}] (* Vladimir Joseph Stephan Orlovsky, May 21 2011 *)
CoefficientList[Series[(1 + 71 x + 111 x^2 + 9 x^3)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 15 2013 *)

Prog

(PARI) a(n)=(2*n+1)*(4*n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [(2n+1)*(4n+1)^2: n in [0..50]]; // G. C. Greubel, Nov 04 2017

Crossrefs

Cf. A002411, A014799, A015224, A014800.

Sequence in context: A350245 A193252 A223452 * A129625 A133382 A199901

Adjacent sequences: A015220 A015221 A015222 * A015224 A015225 A015226

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

More terms from Erich Friedman

Status

approved