cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A158462 a(n) = 36*n^2 - 6.

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%I A158462 #33 Jul 19 2025 16:59:48
%S A158462 30,138,318,570,894,1290,1758,2298,2910,3594,4350,5178,6078,7050,8094,
%T A158462 9210,10398,11658,12990,14394,15870,17418,19038,20730,22494,24330,
%U A158462 26238,28218,30270,32394,34590,36858,39198,41610,44094,46650,49278,51978,54750,57594,60510
%N A158462 a(n) = 36*n^2 - 6.
%C A158462 The identity (12*n^2 - 1)^2 - (36*n^2 - 6)*(2*n)^2 = 1 can be written as A158463(n)^2 - a(n)*A005843(n)^2 = 1.
%H A158462 Vincenzo Librandi, <a href="/A158462/b158462.txt">Table of n, a(n) for n = 1..10000</a>
%H A158462 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158462 G.f.: 6*x*(5 + 8*x - x^2)/(1-x)^3. - _Bruno Berselli_, Aug 27 2011
%F A158462 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Feb 12 2012
%F A158462 From _Amiram Eldar_, Mar 05 2023: (Start)
%F A158462 Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(6))*Pi/sqrt(6))/12.
%F A158462 Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(6))*Pi/sqrt(6) - 1)/12. (End)
%F A158462 From _Elmo R. Oliveira_, Jan 16 2025: (Start)
%F A158462 E.g.f.: 6*(exp(x)*(6*x^2 + 6*x - 1) + 1).
%F A158462 a(n) = 6*A140811(n). (End)
%t A158462 LinearRecurrence[{3, -3, 1}, {30, 138, 318}, 50] (* _Vincenzo Librandi_, Feb 12 2012 *)
%t A158462 36Range[50]^2-6 (* _Harvey P. Dale_, Jul 19 2025 *)
%o A158462 (Magma) I:=[30, 138, 318]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Feb 12 2012
%o A158462 (PARI) for(n=1, 40, print1(36*n^2-6", ")); \\ _Vincenzo Librandi_, Feb 12 2012
%Y A158462 Cf. A005843, A140811, A158463.
%K A158462 nonn,easy
%O A158462 1,1
%A A158462 _Vincenzo Librandi_, Mar 19 2009