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 A041613 #55 Jan 05 2025 19:51:35
%S A041613 1,36,1297,46728,1683505,60652908,2185188193,78727427856,
%T A041613 2836372591009,102188140704180,3681609437941489,132640127906597784,
%U A041613 4778726214075461713,172166783834623219452,6202782944260511361985,223472352777213032250912
%N A041613 Denominators of continued fraction convergents to sqrt(325).
%C A041613 From _Michael A. Allen_, Jul 13 2023: (Start)
%C A041613 Also called the 36-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C A041613 a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 36 kinds of squares available. (End)
%C A041613 a(2*n) and b(2*n) = A041612(2*n) give all (positive integer) solutions to the Pell equation b^2 - 13*a^2 = -1. a(2*n+1) and b(2*n+1) = A041612(2*n+1) give all (positive integer) solutions to the Pell equation b^2 - 13*a^2 = 1. - _Robert FERREOL_, Oct 09 2024
%H A041613 Vincenzo Librandi, <a href="/A041613/b041613.txt">Table of n, a(n) for n = 0..200</a>
%H A041613 Michael A. Allen and Kenneth Edwards, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.
%H A041613 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A041613 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (36,1).
%F A041613 a(n) = F(n, 36), the n-th Fibonacci polynomial evaluated at x=36. - _T. D. Noe_, Jan 19 2006
%F A041613 From _Philippe Deléham_, Nov 23 2008: (Start)
%F A041613 a(n) = 36*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=36.
%F A041613 G.f.: 1/(1 - 36*x - x^2). (End)
%F A041613 a(n) = ((18 + 5*sqrt(13))^(n+1) - (18 - 5*sqrt(13))^(n+1)) / (2*sqrt(13)). - _Robert FERREOL_, Oct 09 2024
%p A041613 with (combinat):seq(fibonacci(3*n,3)/10, n=1..15); # _Zerinvary Lajos_, Apr 20 2008
%t A041613 a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*36,{n,3*4!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Oct 27 2009 *)
%t A041613 Denominator[Convergents[Sqrt[325], 30]] (* _Vincenzo Librandi_ Dec 21 2013 *)
%Y A041613 Cf. A041612 (numerators), A040306 (continued fraction), A295330.
%Y A041613 Row n=36 of A073133, A172236 and A352361 and column k=36 of A157103.
%K A041613 nonn,frac,easy
%O A041613 0,2
%A A041613 _N. J. A. Sloane_
%E A041613 More terms from _Colin Barker_, Nov 20 2013