A158767 - OEIS

%I #25 Mar 23 2023 03:30:47

%S 1,77,305,685,1217,1901,2737,3725,4865,6157,7601,9197,10945,12845,

%T 14897,17101,19457,21965,24625,27437,30401,33517,36785,40205,43777,

%U 47501,51377,55405,59585,63917,68401,73037,77825,82765,87857,93101,98497,104045,109745,115597

%N a(n) = 76*n^2 + 1.

%C The identity (76*n^2 + 1)^2 - (1444*n^2 + 38)*(2*n)^2 = 1 can be written as a(n)^2 - A158766(n)*A005843(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A158767/b158767.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.: -(1 + 74*x + 77*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/(2*sqrt(19)))*Pi/(2*sqrt(19)) + 1)/2.

%F Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(19)))*Pi/(2*sqrt(19)) + 1)/2. (End)

%t LinearRecurrence[{3, -3, 1}, {1, 77, 305}, 50] (* _Vincenzo Librandi_, Feb 21 2012 *)

%t 76*Range[0,40]^2+1 (* _Harvey P. Dale_, Jan 19 2016 *)

%o (Magma) I:=[1, 77, 305]; [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(76*n^2 + 1", ")); \\ _Vincenzo Librandi_, Feb 21 2012

%Y Cf. A005843, A158766.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Mar 26 2009

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