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A158685 a(n) = 32*(32*n^2 + 1).

%I A158685 #26 Jan 15 2025 21:44:00
%S A158685 32,1056,4128,9248,16416,25632,36896,50208,65568,82976,102432,123936,
%T A158685 147488,173088,200736,230432,262176,295968,331808,369696,409632,
%U A158685 451616,495648,541728,589856,640032,692256,746528,802848,861216,921632,984096,1048608,1115168,1183776
%N A158685 a(n) = 32*(32*n^2 + 1).
%C A158685 The identity (64*n^2 + 1)^2 - (1024*n^2 + 32)*(2*n)^2 = 1 can be written as A158686(n)^2 - a(n)*A005843(n)^2 = 1.
%H A158685 Vincenzo Librandi, <a href="/A158685/b158685.txt">Table of n, a(n) for n = 0..1000</a>
%H A158685 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 A158685 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158685 G.f.: -32*(1+30*x+33*x^2)/(x-1)^3.
%F A158685 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158685 From _Amiram Eldar_, Mar 21 2023: (Start)
%F A158685 Sum_{n>=0} 1/a(n) = (coth(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)) + 1)/64.
%F A158685 Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)) + 1)/64. (End)
%F A158685 From _Elmo R. Oliveira_, Jan 15 2025: (Start)
%F A158685 E.g.f.: 32*exp(x)*(1 + 32*x + 32*x^2).
%F A158685 a(n) = 32*A158575(n). (End)
%t A158685 CoefficientList[Series[ - 32 (1 + 30 x + 33 x^2) / (x - 1)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 11 2013 *)
%o A158685 (Magma) [32*(32*n^2+1): n in [0..40]]; // _Vincenzo Librandi_, Sep 11 2013
%o A158685 (PARI) a(n)=32*(32*n^2+1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A158685 Cf. A005843, A158575, A158686.
%K A158685 nonn,easy
%O A158685 0,1
%A A158685 _Vincenzo Librandi_, Mar 24 2009
%E A158685 Comment rewritten, a(0) added and formula replaced by _R. J. Mathar_, Oct 22 2009