A033579 Four times pentagonal numbers: a(n) = 2*n*(3*n-1).

0, 4, 20, 48, 88, 140, 204, 280, 368, 468, 580, 704, 840, 988, 1148, 1320, 1504, 1700, 1908, 2128, 2360, 2604, 2860, 3128, 3408, 3700, 4004, 4320, 4648, 4988, 5340, 5704, 6080, 6468, 6868, 7280, 7704, 8140, 8588, 9048, 9520, 10004, 10500, 11008, 11528, 12060

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Subsequence of A062717: A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011

Sequence found by reading the line from 0, in the direction 0, 4, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Pentagonal Number.
• Wikipedia, Pentagonal number.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000326, A001318, A014642, A033579, A033580, A033581, A049450, A186423.

Programs

GAP

List([0..45], n-> 4*Binomial(3*n,2)/3 ); # G. C. Greubel, Oct 09 2019

Magma

[4*Binomial(3*n,2)/3: n in [0..45]]; // G. C. Greubel, Oct 09 2019

Maple

seq(4*binomial(3*n,2)/3, n=0..45); # G. C. Greubel, Oct 09 2019

Mathematica

4 PolygonalNumber[5, Range[0, 45]] (* Michael De Vlieger, Aug 02 2016, Version 10.4 *)

PARI

a(n)=2*n*(3*n-1) \\ Charles R Greathouse IV, Jun 28 2013

Sage

[4*binomial(3*n,2)/3 for n in (0..45)] # G. C. Greubel, Oct 09 2019

Formula

a(n) = 4*n*(3*n-1)/2 = 6*n^2 - 2*n = 4*A000326(n). - Omar E. Pol, Dec 11 2008

a(n) = 2*A049450(n). - Omar E. Pol, Dec 13 2008

a(n) = a(n-1) + 12*n - 8 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 05 2010

a(n) = A014642(n)/2. - Omar E. Pol, Aug 19 2011

G.f.: x*(4+8*x)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 06 2012

a(n) = A191967(2*n). - Reinhard Zumkeller, Jul 07 2012

a(n) = A181617(n+1) - A181617(n). - J. M. Bergot, Jun 28 2013

a(n) = (A174371(n) - 1)/6. - Miquel Cerda, Jul 28 2016

From Ilya Gutkovskiy, Jul 28 2016: (Start)

E.g.f.: 2*x*(2 + 3*x)*exp(x).

a(n+1) = Sum_{k=0..n} A017569(k).

Sum_{i>0} 1/a(i) = (9*log(3) - sqrt(3)*Pi)/12 = 0.3705093754425278... (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(2*sqrt(3)) - log(2). - Amiram Eldar, Feb 20 2022

Extensions

More terms from Michel Marcus, Mar 04 2014