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 A140811 #95 Jan 16 2025 12:11:18
%S A140811 -1,5,23,53,95,149,215,293,383,485,599,725,863,1013,1175,1349,1535,
%T A140811 1733,1943,2165,2399,2645,2903,3173,3455,3749,4055,4373,4703,5045,
%U A140811 5399,5765,6143,6533,6935,7349,7775,8213,8663,9125,9599,10085,10583,11093,11615
%N A140811 a(n) = 6*n^2 - 1.
%C A140811 Also: The numerators in the j=2 column of the array a(i,j) defined in A140825, where the columns j=0 and j=1 are represented by A000012 and A005408. This could be extended to column j=3: 1, -1, 9, 55, 161, ... The common feature of these sequences derived from a(i,j) is that their j-th differences are constant sequences defined by A091137(j).
%C A140811 a(n) is the set of all k such that 6*k + 6 is a perfect square. - _Gary Detlefs_, Mar 04 2010
%C A140811 The identity (6*n^2 - 1)^2 - (9*n^2 - 3)*(2*n)^2 = 1 can be written as a(n+1)^2 - A157872(n)*A005843(n+1)^2 = 1. - _Vincenzo Librandi_, Feb 05 2012
%C A140811 Apart from first term, sequence found by reading the line from 5, in the direction 5, 23, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - _Omar E. Pol_, Jul 18 2012
%C A140811 From _Paul Curtz_, Sep 17 2018: (Start)
%C A140811 Terms from center to right in the following spiral:
%C A140811 .
%C A140811 65--63--61--59
%C A140811 / \
%C A140811 67 31--29--27 57
%C A140811 / / \ \
%C A140811 69 33 9---7 25 55
%C A140811 / / / \ \ \
%C A140811 71 35 11 -1===5==23==53==>
%C A140811 / / / / / /
%C A140811 37 13 1---3 21 51
%C A140811 \ \ / /
%C A140811 39 15--17--19 49
%C A140811 \ /
%C A140811 41--43--45--47 (End)
%D A140811 P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969, 132 pages, pp. 28-36. CCSA, then CELAR. Now DGA Maitrise de l'Information 35131 Bruz.
%H A140811 Vincenzo Librandi, <a href="/A140811/b140811.txt">Table of n, a(n) for n = 0..1000</a>
%H A140811 Leo Tavares, <a href="/A140811/a140811_1.jpg">Illustration: Barred Stars</a>.
%H A140811 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A140811 a(n) = 2*a(n-1) - a(n-2) + 12.
%F A140811 First differences: a(n+1) - a(n) = A017593(n).
%F A140811 Second differences: A071593(n+1) - A071593(n) = 12.
%F A140811 G.f.: (1-8*x-5*x^2)/(x-1)^3. - _Jaume Oliver Lafont_, Aug 30 2009
%F A140811 From _Vincenzo Librandi_, Feb 05 2012: (Start)
%F A140811 a(n) = a(n-1) + 12*n - 6.
%F A140811 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A140811 a(n) = A033581(n) - 1. - _Omar E. Pol_, Jul 18 2012
%F A140811 a(n) = A032528(2*n) - 1. - _Adriano Caroli_, Jul 21 2013
%F A140811 For n > 0, a(n) = floor(3/(cosh(1/n) - 1)) = floor(1/(n*sinh(1/n) - 1)); for similar formulas for cosine and sine, see A033581. - _Clark Kimberling_, Oct 19 2014, corrected by _M. F. Hasler_, Oct 21 2014
%F A140811 a(-n) = a(n). - _Paul Curtz_, Sep 17 2018
%F A140811 From _Amiram Eldar_, Feb 04 2021: (Start)
%F A140811 Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(6))*cot(Pi/sqrt(6)))/2.
%F A140811 Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(6))*csc(Pi/sqrt(6)) - 1)/2.
%F A140811 Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(6))*csc(Pi/sqrt(6)).
%F A140811 Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(6))*sin(Pi/sqrt(3))/sqrt(2). (End)
%F A140811 a(n) = A003154(n+1) - 2*A016777(n). - _Leo Tavares_, May 13 2022
%F A140811 E.g.f.: exp(x)*(6*x^2 + 6*x - 1). - _Elmo R. Oliveira_, Jan 16 2025
%t A140811 LinearRecurrence[{3,-3,1},{-1,5,23},40] (* _Vincenzo Librandi_, Feb 05 2012 *)
%t A140811 CoefficientList[Series[(1-8*x-5*x^2)/(x-1)^3 , {x, 0, 40}], x] (* _Stefano Spezia_, Sep 17 2018 *)
%o A140811 (PARI) a(n)=6*n^2-1 \\ _Charles R Greathouse IV_, Jun 01 2011
%o A140811 (Magma) [6*n^2 - 1: n in [0..50]]; // _Vincenzo Librandi_, Jun 02 2011
%Y A140811 Cf. A005843, A157872, A060747, A103115(n+1), A141417 (array).
%Y A140811 Cf. A000012, A001318, A005408, A091137, A140825.
%Y A140811 Cf. A003154, A032528, A033581, A016777, A017593.
%K A140811 sign,easy
%O A140811 0,2
%A A140811 _Paul Curtz_, Jul 16 2008
%E A140811 Edited and extended by _R. J. Mathar_, Aug 06 2008
%E A140811 Better description _Ray Chandler_, Feb 03 2009