A042013 - OEIS

%I #38 Jan 05 2025 19:51:35

%S 1,46,2117,97428,4483805,206352458,9496696873,437054408616,

%T 20113999493209,925681031096230,42601441429919789,1960591986807406524,

%U 90229832834570619893,4152532902377055921602,191106743342179143013585,8795062726642617634546512

%N Denominators of continued fraction convergents to sqrt(530).

%C From _Michael A. Allen_, Dec 02 2023: (Start)

%C Also called the 46-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 46 kinds of squares available. (End)

%H Vincenzo Librandi, <a href="/A042013/b042013.txt">Table of n, a(n) for n = 0..200</a>

%H 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 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 (46,1).

%F a(n) = F(n, 46), the n-th Fibonacci polynomial evaluated at x=46. - _T. D. Noe_, Jan 19 2006

%F From _Philippe Deléham_, Nov 23 2008: (Start)

%F a(n) = 46*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=46.

%F G.f.: 1/(1-46*x-x^2). (End)

%t Denominator[Convergents[Sqrt[530], 40]] (* _Vincenzo Librandi_, Jan 12 2014 *)

%Y Cf. A042012, A040506.

%Y Row n=46 of A073133, A172236 and A352361 and column k=46 of A157103.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_

%E Additional term from _Colin Barker_, Nov 29 2013