A033580 Four times second pentagonal numbers: a(n) = 2*n*(3*n+1).

0, 8, 28, 60, 104, 160, 228, 308, 400, 504, 620, 748, 888, 1040, 1204, 1380, 1568, 1768, 1980, 2204, 2440, 2688, 2948, 3220, 3504, 3800, 4108, 4428, 4760, 5104, 5460, 5828, 6208, 6600, 7004, 7420, 7848, 8288, 8740, 9204, 9680, 10168, 10668, 11180, 11704, 12240

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, 8,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A139267 in the same spiral - Omar E. Pol, Sep 09 2011

a(n) is the number of edges of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - Emeric Deutsch May 13 2018

The partial sums of this sequence give A035006. - Leo Tavares, Oct 03 2021

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, Computing topological indices of certain networks, J. of Optoelectronics and Advanced Materials, 18, No. 9-10 (2016), pp. 884-892.
• Leo Tavares, Illustration: Crossed Stars
• Leo Tavares, Illustration: Four Quarter Star Crosses
• Leo Tavares, Illustration: Triangulated Star Crosses
• Leo Tavares, Illustration: Oblong Star Crosses
• Leo Tavares, Illustration: Crossed Diamond Stars
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A033579, A049451, A045945, A186423.

Cf. sequences listed in A254963.

Cf. A003154, A016813.

Cf. A035006.

Cf. A005449, A046092, A033996, A002378, A016742, A000290, A016754, A001105.

Programs

GAP

List([0..50], n-> 2*n*(3*n+1)); # G. C. Greubel, Oct 09 2019

Magma

[2*n*(3*n+1): n in [0..50]]; // G. C. Greubel, Oct 09 2019

Maple

seq(2*n*(3*n+1), n=0..50); # G. C. Greubel, Oct 09 2019

Mathematica

4*Binomial[3*Range[50]-2, 2]/3 (* G. C. Greubel, Oct 09 2019 *)

PARI

a(n)=2*n*(3*n+1) \\ Charles R Greathouse IV, Sep 28 2015

Sage

[2*n*(3*n+1) for n in (0..50)] # G. C. Greubel, Oct 09 2019

Formula

a(n) = a(n-1) +12*n -4 (with a(0)=0). - Vincenzo Librandi, Aug 05 2010

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

a(-n) = A033579(n). - Michael Somos, Jun 09 2014

E.g.f.: 2*x*(4 + 3*x)*exp(x). - G. C. Greubel, Oct 09 2019

From Amiram Eldar, Jan 14 2021: (Start)

Sum_{n>=1} 1/a(n) = 3/2 - Pi/(4*sqrt(3)) - 3*log(3)/4.

Sum_{n>=1} (-1)^(n+1)/a(n) = -3/2 + Pi/(2*sqrt(3)) + log(2). (End)

From Leo Tavares, Oct 12 2021: (Start)

a(n) = A003154(n+1) - A016813(n). See Crossed Stars illustration.

a(n) = 4*A005449(n). See Four Quarter Star Crosses illustration.

a(n) = 2*A049451(n).

a(n) = A046092(n-1) + A033996(n). See Triangulated Star Crosses illustration.

a(n) = 4*A000217(n-1) + 8*A000217(n).

a(n) = 4*A000217(n-1) + 4*A002378. See Oblong Star Crosses illustration.

a(n) = A016754(n) + 4*A000217(n). See Crossed Diamond Stars illustration.

a(n) = 2*A001105(n) + 4*A000217(n).

a(n) = A016742(n) + A046092(n).

a(n) = 4*A000290(n) + 4*A000217(n). (End)