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%I #41 Feb 26 2022 09:19:12
%S 15,36,63,96,135,180,231,288,351,420,495,576,663,756,855,960,1071,
%T 1188,1311,1440,1575,1716,1863,2016,2175,2340,2511,2688,2871,3060,
%U 3255,3456,3663,3876,4095,4320,4551,4788,5031,5280,5535,5796,6063,6336,6615,6900
%N a(n) = 3*n^2 + 12*n.
%C Numbers k such that 12*(12 + k) is a perfect square.
%C 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
%H Vincenzo Librandi, <a href="/A067707/b067707.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: 3*x*(5 - 3*x)/(1 - x)^3. - _Vincenzo Librandi_, Jul 07 2012
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Jul 07 2012
%F E.g.f.: 3*x*(x + 5)*exp(x). - _G. C. Greubel_, Jul 20 2017
%F From _Amiram Eldar_, Feb 26 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 25/144.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 7/144. (End)
%t Select[ Range[10000], IntegerQ[ Sqrt[ 12(12 + # )]] & ]
%t CoefficientList[Series[3*(5-3*x)/(1-x)^3,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 07 2012 *)
%o (PARI) a(n)=3*n*(n+4) \\ _Charles R Greathouse IV_, Dec 07 2011
%o (Magma) [3*n^2 + 12*n: n in [1..50]]; // _Vincenzo Librandi_, Jul 07 2012
%Y Cf. A067724 (5), A067725 (3), A067726 (6), A067727 (7), A067728, A067705 (11).
%K nonn,easy
%O 1,1
%A _Robert G. Wilson v_, Feb 05 2002
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