A067707 a(n) = 3*n^2 + 12*n.

15, 36, 63, 96, 135, 180, 231, 288, 351, 420, 495, 576, 663, 756, 855, 960, 1071, 1188, 1311, 1440, 1575, 1716, 1863, 2016, 2175, 2340, 2511, 2688, 2871, 3060, 3255, 3456, 3663, 3876, 4095, 4320, 4551, 4788, 5031, 5280, 5535, 5796, 6063, 6336, 6615, 6900

Offset

1,1

Comments

Numbers k such that 12*(12 + k) is a perfect square.

a(n) is the second Zagreb index of the gear graph g[n]. The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph. The gear graph g[n] is defined as a wheel graph with n+1 vertices with a vertex added between each pair of adjacent vertices of the outer cycle. - Emeric Deutsch, Nov 09 2016

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Gear Graph.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: 3*x*(5 - 3*x)/(1 - x)^3. - Vincenzo Librandi, Jul 07 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 07 2012

E.g.f.: 3*x*(x + 5)*exp(x). - G. C. Greubel, Jul 20 2017

From Amiram Eldar, Feb 26 2022: (Start)

Sum_{n>=1} 1/a(n) = 25/144.

Sum_{n>=1} (-1)^(n+1)/a(n) = 7/144. (End)

Mathematica

Select[ Range[10000], IntegerQ[ Sqrt[ 12(12 + # )]] & ]
CoefficientList[Series[3*(5-3*x)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 07 2012 *)

Prog

(PARI) a(n)=3*n*(n+4) \\ Charles R Greathouse IV, Dec 07 2011
(Magma) [3*n^2 + 12*n: n in [1..50]]; // Vincenzo Librandi, Jul 07 2012

Crossrefs

Cf. A067724 (5), A067725 (3), A067726 (6), A067727 (7), A067728, A067705 (11).

Sequence in context: A062712 A224719 A033709 * A166146 A229235 A346881

Adjacent sequences: A067704 A067705 A067706 * A067708 A067709 A067710

Keyword

nonn,easy

Author

Robert G. Wilson v, Feb 05 2002

Status

approved