A158628 - OEIS

%I #25 Mar 16 2023 04:03:37

%S 43,175,395,703,1099,1583,2155,2815,3563,4399,5323,6335,7435,8623,

%T 9899,11263,12715,14255,15883,17599,19403,21295,23275,25343,27499,

%U 29743,32075,34495,37003,39599,42283,45055,47915,50863,53899,57023,60235,63535,66923,70399

%N a(n) = 44*n^2 - 1.

%C The identity (44*n^2 - 1)^2 - (484*n^2 - 22)*(2*n)^2 = 1 can be written as a(n)^2 - A158627(n)*A005843(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A158628/b158628.txt">Table of n, a(n) for n = 1..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.: x*(-43 - 46*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 16 2023: (Start)

%F Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(11)))*Pi/(2*sqrt(11)))/2.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(11)))*Pi/(2*sqrt(11)) - 1)/2. (End)

%t 44Range[0,40]^2-1 (* or *) CoefficientList[Series[(1-46 x-43 x^2)/ (x-1)^3,{x,0,40}],x] (* _Harvey P. Dale_, Apr 22 2011 *)

%t LinearRecurrence[{3, -3, 1}, {43, 175, 395}, 40] (* _Vincenzo Librandi_, Feb 17 2012 *)

%o (Magma) I:=[43, 175, 395]; [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 17 2012

%o (PARI) for(n=1, 40, print1(44*n^2-1", ")); \\ _Vincenzo Librandi_, Feb 17 2012

%Y Cf. A005843, A158627.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 23 2009

%E Comment rewritten, formula replaced by _R. J. Mathar_, Oct 28 2009