A042513 - OEIS

%I #39 Jan 05 2025 19:51:35

%S 1,56,3137,175728,9843905,551434408,30890170753,1730400996576,

%T 96933345979009,5429997775821080,304176808791959489,

%U 17039331290125552464,954506729055822897473,53469416158416207810952,2995241811600363460310785,167787010865778769985214912

%N Denominators of continued fraction convergents to sqrt(785).

%C From _Michael A. Allen_, Dec 17 2023: (Start)

%C Also called the 56-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 56 kinds of squares available. (End)

%H Vincenzo Librandi, <a href="/A042513/b042513.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 (56,1).

%F a(n) = F(n, 56), the n-th Fibonacci polynomial evaluated at x=56. - _T. D. Noe_, Jan 19 2006

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

%F a(n) = 56*a(n-1) + a(n-2) for n > 1, a(0)=1, a(1)=56.

%F G.f.: 1/(1 - 56*x - x^2). (End)

%t a=0; lst={}; s=0; Do[a = s-(a-1); AppendTo[lst, a]; s+=a*56, {n, 3*4!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)

%t Denominator[Convergents[Sqrt[785], 30]] (* _Harvey P. Dale_, Jun 26 2012 *)

%t CoefficientList[Series[1/(1 - 56 x - x^2), {x, 0, 25}], x] (* _Vincenzo Librandi_, Jan 23 2014 *)

%Y Cf. A042512, A040756.

%Y Row n=56 of A073133, A172236 and A352361 and column k=56 of A157103.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_

%E Additional term from _Colin Barker_, Dec 17 2013