This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A114047 #54 Sep 21 2024 13:25:02
%S A114047 1,649,842401,1093435849,1419278889601,1842222905266249,
%T A114047 2391203911756701601,3103780835237293411849,4028705132934095091878401,
%U A114047 5229256158767620191964752649,6787570465375238075075157060001,8810261234800900253827361899128649
%N A114047 x such that x^2 - 13*y^2 = 1.
%C A114047 A Pellian equation (Pell's equation). - _Benoit Cloitre_, Feb 03 2006
%C A114047 Numbers n such that 13*(n^2-1) is a square. - _Vincenzo Librandi_, Nov 13 2010
%C A114047 The corresponding values y of the solutions of this Pell equation are given in A075871(n). - _Wolfdieter Lang_, Jun 27 2013
%H A114047 Colin Barker, <a href="/A114047/b114047.txt">Table of n, a(n) for n = 0..321</a>
%H A114047 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A114047 John Robertson, <a href="https://web.archive.org/web/20190501092519/http://www.jpr2718.org/">Home page</a>.
%H A114047 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1298,-1).
%F A114047 a(0)=1, a(1)=649 then a(n)=1298*a(n-1)-a(n-2). - _Benoit Cloitre_, Feb 03 2006
%F A114047 G.f.: (1-649*x)/(1-1298*x+x^2). - _Philippe Deléham_, Nov 18 2008
%F A114047 a(n) = 2*A132644(n) + 1. - _Hugo Pfoertner_, Feb 11 2024
%e A114047 (649^2-1)/13 = 180^2.
%t A114047 LinearRecurrence[{1298,-1},{1,649},20] (* or *) With[{c=180Sqrt[13]}, Simplify[Table[1/2((649-c)^n+(649+c)^n),{n,0,20}]]] (* _Harvey P. Dale_, Aug 11 2011 *)
%o A114047 (PARI) /* This sequence is computed with g(1e9,13) in the following program. */
%o A114047 g(n,k) = for(y=0,n,x=k*y^2+1;if(issquare(x),print1(floor(sqrt(x))",")))
%o A114047 (PARI) a0=1;a1=649;for(n=2,30,a2=1298*a1-a0;a0=a1;a1=a2;print1(a2,",")) \\ _Benoit Cloitre_
%o A114047 (PARI) Vec((1-649*x)/(1-1298*x+x^2) + O(x^100)) \\ _Colin Barker_, Jun 13 2015
%o A114047 (Magma) I:=[1,649]; [n le 2 select I[n] else 1298*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Jun 14 2015
%Y A114047 Cf. A202155, A075871, A132644.
%K A114047 nonn,easy
%O A114047 0,2
%A A114047 _Cino Hilliard_, Feb 01 2006
%E A114047 More terms from _Benoit Cloitre_, Feb 03 2006