A158741 - OEIS

%I #23 Mar 23 2023 03:29:25

%S 37,1406,5513,12358,21941,34262,49321,67118,87653,110926,136937,

%T 165686,197173,231398,268361,308062,350501,395678,443593,494246,

%U 547637,603766,662633,724238,788581,855662,925481,998038,1073333,1151366,1232137,1315646,1401893,1490878

%N a(n) = 1369*n^2 + 37.

%C The identity (74*n^2 + 1)^2 - (1369*n^2 + 37)*(2*n)^2 = 1 can be written as A158742(n)^2 - a(n)*A005843(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A158741/b158741.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: -37*(1 + 35*x + 38*x^2)/(x-1)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F From _Amiram Eldar_, Mar 23 2023: (Start)

%F Sum_{n>=0} 1/a(n) = (coth(Pi/sqrt(37))*Pi/sqrt(37) + 1)/74.

%F Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/sqrt(37))*Pi/sqrt(37) + 1)/74. (End)

%t LinearRecurrence[{3, -3, 1}, {37, 1406, 5513}, 50] (* _Vincenzo Librandi_, Feb 21 2012 *)

%o (Magma) I:=[37, 1406, 5513]; [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 21 2012

%o (PARI) for(n=0, 40, print1(1369*n^2 + 37", ")); \\ _Vincenzo Librandi_, Feb 21 2012

%Y Cf. A005843, A158742.

%K nonn,easy

%O 0,1

%A _Vincenzo Librandi_, Mar 25 2009

%E Comment rewritten, a(0) added and formula replaced by _R. J. Mathar_, Oct 22 2009