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%I #30 Mar 06 2023 02:21:28
%S 156,663,1508,2691,4212,6071,8268,10803,13676,16887,20436,24323,28548,
%T 33111,38012,43251,48828,54743,60996,67587,74516,81783,89388,97331,
%U 105612,114231,123188,132483,142116,152087,162396,173043,184028,195351,207012,219011,231348
%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 A158551(n)^2 - a(n)*A005843(n)^2 = 1.
%H Vincenzo Librandi, <a href="/A158550/b158550.txt">Table of n, a(n) for n = 1..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*x*(-12 - 15*x + x^2)/(x-1)^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>=1} 1/a(n) = (1 - cot(Pi/sqrt(13))*Pi/sqrt(13))/26.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(13))*Pi/sqrt(13) - 1)/26. (End)
%t LinearRecurrence[{3, -3, 1}, {156, 663, 1508}, 40] (* _Vincenzo Librandi_, Feb 14 2012 *)
%t 169*Range[40]^2-13 (* _Harvey P. Dale_, Apr 12 2017 *)
%o (Magma) I:=[156, 663, 1508]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Feb 14 2012
%o (PARI) for(n=1, 40, print1(169*n^2 - 13", ")); \\ _Vincenzo Librandi_, Feb 14 2012
%Y Cf. A005843, A158551.
%K nonn,easy
%O 1,1
%A _Vincenzo Librandi_, Mar 21 2009
%E Comment rewritten by _R. J. Mathar_, Oct 16 2009
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