A049598 12 times triangular numbers.

0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792, 936, 1092, 1260, 1440, 1632, 1836, 2052, 2280, 2520, 2772, 3036, 3312, 3600, 3900, 4212, 4536, 4872, 5220, 5580, 5952, 6336, 6732, 7140, 7560, 7992, 8436, 8892, 9360, 9840, 10332, 10836, 11352

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

Comments

a(n-1) is the Wiener index of the helm graph H(n) (n>=3). The graph H(n) is obtained from an n-wheel graph (on n+1 nodes) by adjoining a pendant edge at each node of the cycle. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. The Wiener polynomial of H(n) is (1/2)*n*t*((n-3)t^3 + 2(n-2)t^2 + (n+3)t + 6). - Emeric Deutsch, Sep 28 2010

Also sequence found by reading the line from 0, in the direction 0, 12, ..., and the same line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. Axis perpendicular to A195158 in the same spiral. - Omar E. Pol, Sep 29 2011

Also the Wiener index of the (n+1)-gear graph. - Eric W. Weisstein, Sep 08 2017

Examples

			a(1) = 12*1 + 0 = 12;
a(2) = 12*2 + 12 = 36;
a(3) = 12*3 + 36 = 72.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., Vol. 60 (1996), pp. 959-969.
• Leo Tavares, Illustration: Centroid Stars.
• Eric Weisstein's World of Mathematics, Gear Graph.
• Eric Weisstein's World of Mathematics, Helm Graph.
• Eric Weisstein's World of Mathematics, Wiener Index.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A001844, A003154, A027468, A035008, A073577.

Programs

Mathematica

12 * Accumulate[Range[0, 50]] (* Harvey P. Dale, Feb 05 2013 *)
(* Start from Eric W. Weisstein, Sep 08 2017 *)
Table[6 n (n + 1), {n, 0, 20}]
12 PolygonalNumber[3, Range[0, 20]]
12 Binomial[Range[20], 2]
LinearRecurrence[{3, -3, 1}, {12, 36, 72}, {0, 20}]
(* End *)

PARI

a(n)=6*n*(n+1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 6*n*(n+1).

G.f.: 12*x/(1-x)^3.

a(n) = 12*A000217(n). - Omar E. Pol, Dec 11 2008

a(n) = 12*n + a(n-1) (with a(0)=0). - Vincenzo Librandi, Aug 06 2010

a(n) = A003154(n+1) - 1. - Omar E. Pol, Oct 03 2011

a(n) = A032528(2*n+1) - 1. - Adriano Caroli, Jul 19 2013

a(n) = A001844(n) + A073577(n). - Bruce J. Nicholson, Aug 06 2017

E.g.f.: 6*x*(x+2)*exp(x). - G. C. Greubel, Aug 23 2017

From Amiram Eldar, Feb 15 2022: (Start)

Sum_{n>=1} 1/a(n) = 1/6.

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/3 - 1/6. (End)

From Amiram Eldar, Feb 21 2023: (Start)

Product_{n>=1} (1 - 1/a(n)) = -(6/Pi)*cos(sqrt(5/3)*Pi/2).

Product_{n>=1} (1 + 1/a(n)) = (6/Pi)*cos(Pi/(2*sqrt(3))). (End)