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%I #28 Mar 06 2023 02:21:36
%S 13,182,689,1534,2717,4238,6097,8294,10829,13702,16913,20462,24349,
%T 28574,33137,38038,43277,48854,54769,61022,67613,74542,81809,89414,
%U 97357,105638,114257,123214,132509,142142,152113,162422,173069,184054,195377,207038,219037
%N a(n) = 169*n^2 + 13.
%C The identity (26*n^2 + 1)^2 - (169*n^2 + 13)*(2*n)^2 = 1 can be written as A158549(n)^2 - a(n)*A005843(n)^2 = 1.
%H Vincenzo Librandi, <a href="/A158548/b158548.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f: 13*(1 + 11*x + 14*x^2)/(1-x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F From _Amiram Eldar_, Mar 06 2023: (Start)
%F Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(13))*Pi/sqrt(13) + 1)/26.
%F Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(13))*Pi/sqrt(13) + 1)/26. (End)
%t LinearRecurrence[{3, -3, 1}, {13, 182, 689}, 50] (* _Vincenzo Librandi_, Feb 14 2012 *)
%o (Magma) I:=[13, 182, 689]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 14 2012
%o (PARI) for(n=1, 40, print1(169*n^2 + 13", ")); \\ _Vincenzo Librandi_, Feb 14 2012
%Y Cf. A158549, A005843.
%K nonn,easy
%O 0,1
%A _Vincenzo Librandi_, Mar 21 2009
%E Comment rewritten, a(0) added by _R. J. Mathar_, Oct 16 2009
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