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 A202156 #30 Feb 11 2024 06:50:22
%S A202156 5,6485,8417525,10925940965,14181862955045,18408047189707445,
%T A202156 23893631070377308565,31013914721302556809925,
%U A202156 40256037414619648361974085,52252305550261582271285552405,67823452348202119168480285047605,88034788895660800419105138706238885
%N A202156 y-values in the solution to x^2 - 13*y^2 = -1.
%C A202156 The corresponding values of x of this Pell equation are in A202155.
%D A202156 A. H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover Publications (New York), 1966, p. 264.
%H A202156 Bruno Berselli, <a href="/A202156/b202156.txt">Table of n, a(n) for n = 1..200</a>
%H A202156 A. J. C. Cunningham, <a href="https://archive.org/details/binomialfactoris01cunn/page/n46/mode/1up">Binomial Factorisations</a>, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
%H A202156 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A202156 A. M. S. Ramasamy, <a href="http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20006851_577.pdf">Polynomial solutions for the Pell's equation</a>, Indian Journal of Pure and Applied Mathematics 25 (1994), p. 579 (Theorem 4, case t=1).
%H A202156 J. P. Robertson, <a href="https://web.archive.org/web/20180831180333/http://www.jpr2718.org/pell.pdf">Solving the generalized Pell equation x^2-D*y^2=N</a>, pp. 9, 24.
%H A202156 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1298,-1).
%F A202156 G.f.: 5*x*(1-x)/(1-1298*x+x^2).
%F A202156 a(n) = a(-n+1) = 5*(r^(2n-1)+1/r^(2n-1))/(r+1/r), where r=18+5*sqrt(13).
%F A202156 a(n) = A006191(6*n - 3). - _Michael Somos_, Feb 24 2023
%t A202156 LinearRecurrence[{1298, -1}, {5, 6485}, 12]
%o A202156 (Magma) m:=13; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(5*x*(1-x)/(1-1298*x+x^2)));
%o A202156 (Maxima) makelist(expand(((18+5*sqrt(13))^(2*n-1)-(18-5*sqrt(13))^(2*n-1))/(2*sqrt(13))), n, 1, 12);
%Y A202156 Cf. A002313, A003654, A006191, A031396, A075871, A202155.
%K A202156 nonn,easy
%O A202156 1,1
%A A202156 _Bruno Berselli_, Dec 15 2011