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
KEYWORD
nonn,easy
AUTHOR
Robert G. Wilson v, Feb 05 2002
STATUS
approved