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.
%I A158463 #72 Aug 10 2025 06:01:21
%S A158463 -1,11,47,107,191,299,431,587,767,971,1199,1451,1727,2027,2351,2699,
%T A158463 3071,3467,3887,4331,4799,5291,5807,6347,6911,7499,8111,8747,9407,
%U A158463 10091,10799,11531,12287,13067,13871,14699,15551,16427,17327,18251,19199,20171,21167,22187
%N A158463 a(n) = 12*n^2 - 1.
%C A158463 Related to Legendre polynomials - see Mma line. - _N. J. A. Sloane_, Nov 17 2009
%C A158463 One notices that this sequence produces an inordinate number of semiprimes, perhaps better than mere chance for large values of n. - _J. M. Bergot_, Jun 30 2011
%C A158463 Sequence found by reading the line from -1, in the direction -1, 11, ..., in the square spiral whose vertices are -1 together with the generalized octagonal numbers A001082. - _Omar E. Pol_, Jul 18 2012
%H A158463 Vincenzo Librandi, <a href="/A158463/b158463.txt">Table of n, a(n) for n = 0..1000</a>
%H A158463 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158463 a(-n) = a(n).
%F A158463 G.f.: (1-14*x-11*x^2)/(x-1)^3. - _R. J. Mathar_, Aug 27 2011
%F A158463 a(n) = A135453(n) - 1. - _Omar E. Pol_, Jul 18 2012
%F A158463 From _Amiram Eldar_, Feb 04 2021: (Start)
%F A158463 Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(12))*cot(Pi/sqrt(12)))/2.
%F A158463 Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(12))*csc(Pi/sqrt(12)) - 1)/2.
%F A158463 Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(12))*csc(Pi/sqrt(12)).
%F A158463 Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(12))*sin(Pi/sqrt(6))/sqrt(2). (End)
%F A158463 From _Gerry Martens_, Apr 06 2024: (Start)
%F A158463 a(n) = Re((2*n*i-1)^3).
%F A158463 a(n) = -8*(1/4+n^2)^(3/2)*cos(3*arctan(2*n)). (End)
%F A158463 From _Elmo R. Oliveira_, Jan 16 2025: (Start)
%F A158463 E.g.f.: exp(x)*(12*x^2 + 12*x - 1).
%F A158463 a(n) = A080663(2*n) for n > 0. (End)
%e A158463 G.f. = -1 + 11*x + 47*x^2 + 107*x^3 + 191*x^4 + 299*x^5 + 431*x^6 + 587*x^7 + 767*x^8 + ...
%t A158463 Table[Numerator[LegendreP[2,2n]],{n,0,50}] (* _N. J. A. Sloane_, Nov 17 2009 *)
%t A158463 a[ n_] := 2 LegendreP[ 2, 2 n] (* _Michael Somos_, Jul 01 2011 *)
%t A158463 12*Range[0,40]^2-1 (* or *) LinearRecurrence[{3,-3,1},{-1,11,47},50] (* _Harvey P. Dale_, Jun 22 2019 *)
%o A158463 (PARI) {a(n) = 12 * n^2 - 1} /* _Michael Somos_, Nov 12 2011 */
%o A158463 (Magma) [12*n^2 - 1: n in [0..100]]; // _G. C. Greubel_, Sep 25 2018
%Y A158463 Cf. A001082, A080663, A135453.
%K A158463 sign,easy
%O A158463 0,2
%A A158463 _Vincenzo Librandi_, Mar 19 2009
%E A158463 Minor edits by _N. J. A. Sloane_, Nov 17 2009
%E A158463 Comment rewritten by _Bruno Berselli_, Aug 27 2011