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A158444 a(n) = 16*n^2 + 4.

%I A158444 #47 Jan 28 2025 11:22:25
%S A158444 20,68,148,260,404,580,788,1028,1300,1604,1940,2308,2708,3140,3604,
%T A158444 4100,4628,5188,5780,6404,7060,7748,8468,9220,10004,10820,11668,12548,
%U A158444 13460,14404,15380,16388,17428,18500,19604,20740,21908,23108,24340,25604,26900,28228
%N A158444 a(n) = 16*n^2 + 4.
%C A158444 The identity (8*n^2 + 1)^2 - (16*n^2 + 4)*(2*n)^2 = 1 can be written as A081585(n)^2 - a(n)*A005843(n)^2 = 1. [rewritten by _Bruno Berselli_, Sep 06 2011]
%C A158444 Sequence found by reading the line from 20, in the direction 20, 68, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - _Omar E. Pol_, Nov 02 2012
%H A158444 Vincenzo Librandi, <a href="/A158444/b158444.txt">Table of n, a(n) for n = 1..10000</a>
%H A158444 Vincenzo Librandi, <a href="https://web.archive.org/web/20090309225914/http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>, Math Forum, 2007. [Wayback Machine link]
%H A158444 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158444 From _Bruno Berselli_, Sep 06 2011: (Start)
%F A158444 G.f.: 4*x*(5 + 2*x + x^2)/(1-x)^3.
%F A158444 a(n) = 4*A053755(n). (End)
%F A158444 From _Amiram Eldar_, Mar 05 2023: (Start)
%F A158444 Sum_{n>=1} 1/a(n) = (coth(Pi/2)*Pi/2 - 1)/8.
%F A158444 Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/2)*Pi/2)/8. (End)
%F A158444 E.g.f.: 4*(exp(x)*(4*x^2 + 4*x + 1) - 1). - _Elmo R. Oliveira_, Jan 27 2025
%t A158444 a[n_] := 16*n^2 + 4; Array[a, 50] (* _Amiram Eldar_, Mar 05 2023 *)
%o A158444 (Magma) [16*n^2+4: n in [1..50]];
%o A158444 (PARI) a(n)=16*n^2+4 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A158444 Cf. A005843, A053755, A074377, A081585.
%K A158444 nonn,easy
%O A158444 1,1
%A A158444 _Vincenzo Librandi_, Mar 19 2009