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%I #36 Jan 22 2024 06:08:49
%S 1,52,2705,140712,7319729,380766620,19807183969,1030354333008,
%T 53598232500385,2788138444353028,145036797338857841,
%U 7544701600064960760,392469520000716817361,20415959741637339463532,1062022376085142368921025,55245579516169040523356832
%N Denominators of continued fraction convergents to sqrt(677).
%C From _Michael A. Allen_, Dec 17 2023: (Start)
%C Also called the 52-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C 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 52 kinds of squares available. (End)
%H Vincenzo Librandi, <a href="/A042301/b042301.txt">Table of n, a(n) for n = 0..200</a>
%H Michael A. Allen and Kenneth Edwards, <a href="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 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (52,1).
%F a(n) = F(n, 52), the n-th Fibonacci polynomial evaluated at x=52. - _T. D. Noe_, Jan 19 2006
%F From _Philippe Deléham_, Nov 23 2008: (Start)
%F a(n) = 52*a(n-1) + a(n-2) for n > 1, a(0)=1, a(1)=52.
%F G.f.: 1/(1 - 52*x - x^2). (End)
%t a = 0; lst = {}; s = 0; Do[a = s - (a - 1); AppendTo[lst, a]; s += a*52, {n, 3*4!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)
%t Denominator[Convergents[Sqrt[677], 30]] (* _Vincenzo Librandi_, Jan 19 2014 *)
%t LinearRecurrence[{52,1},{1,52},20] (* _Harvey P. Dale_, Mar 24 2023 *)
%Y Cf. A042300, A040650.
%Y Row n=52 of A073133, A172236 and A352361 and column k=52 of A157103.
%K nonn,frac,easy
%O 0,2
%A _N. J. A. Sloane_
%E Additional term from _Colin Barker_, Dec 07 2013
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