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A158480 a(n) = 12*n^2 + 1.

%I A158480 #67 Sep 08 2022 08:45:43
%S A158480 1,13,49,109,193,301,433,589,769,973,1201,1453,1729,2029,2353,2701,
%T A158480 3073,3469,3889,4333,4801,5293,5809,6349,6913,7501,8113,8749,9409,
%U A158480 10093,10801,11533,12289,13069,13873,14701,15553,16429,17329,18253,19201,20173,21169
%N A158480 a(n) = 12*n^2 + 1.
%C A158480 The identity (12*n^2 + 1)^2 - (36*n^2 + 6)*(2*n)^2 = 1 can be written as a(n)^2 - A158479(n)*A005843(n)^2 = 1.
%C A158480 Sequence found by reading the line from 13, in the direction 13, 49, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - _Omar E. Pol_, Jul 18 2012
%H A158480 Vincenzo Librandi, <a href="/A158480/b158480.txt">Table of n, a(n) for n = 0..10000</a>
%H A158480 Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a> [broken link]
%H A158480 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158480 a(n) = A010014(n)/2. - _Vladimir Joseph Stephan Orlovsky_, May 18 2009
%F A158480 G.f: (13*x^2 + 10*x + 1)/(1-x)^3.
%F A158480 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158480 a(n) = 1 + A135453(n). - _Omar E. Pol_, Jul 18 2012
%F A158480 a(n) = (A016969(n-1)*A016921(n) + 4)/3. - _Hilko Koning_, Oct 25 2019
%F A158480 E.g.f.: (1 + 12*x + 12*x^2)*exp(x). - _G. C. Greubel_, Nov 06 2019
%F A158480 From _Amiram Eldar_, Jul 15 2020: (Start)
%F A158480 Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(12))*coth(Pi/sqrt(12)))/2.
%F A158480 Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(12))*csch(Pi/sqrt(12)))/2. (End)
%F A158480 From _Amiram Eldar_, Feb 05 2021: (Start)
%F A158480 Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(12))*sinh(Pi/sqrt(6)).
%F A158480 Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(12))*csch(Pi/sqrt(12)). (End)
%e A158480 a(1) = 12*1^2 + 1 = 13.
%e A158480 a(2) = 12*2^2 + 1 = 49.
%e A158480 a(3) = 12*3^2 + 1 = 109.
%p A158480 seq(12*n^2 +1, n=0..45); # _G. C. Greubel_, Nov 06 2019
%t A158480 LinearRecurrence[{3,-3,1}, {13,49,109}, 40]
%t A158480 12*Range[40]^2 +1 (* _G. C. Greubel_, Nov 06 2019 *)
%o A158480 (Magma) I:=[13,49,109];[n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2) +Self(n-3): n in [1..50]];
%o A158480 (PARI) a(n)=12*n^2+1
%o A158480 (Sage) [12*n^2 +1 for n in (1..40)] # _G. C. Greubel_, Nov 06 2019
%o A158480 (GAP) List([1..40], n-> 12*n^2 + 1); # _G. C. Greubel_, Nov 06 2019
%Y A158480 Cf. A005843, A158479.
%K A158480 nonn,easy
%O A158480 0,2
%A A158480 _Vincenzo Librandi_, Mar 20 2009
%E A158480 a(0)=1 prepended by _Alois P. Heinz_, Jun 12 2021