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A042405 Denominators of continued fraction convergents to sqrt(730). 3

%I #47 Jan 05 2024 08:04:42

%S 1,54,2917,157572,8511805,459795042,24837444073,1341681774984,

%T 72475653293209,3915026959608270,211483931472139789,

%U 11424047326455156876,617110039560050611093,33335366183569188155898,1800726883952296211029585,97272587099607564583753488

%N Denominators of continued fraction convergents to sqrt(730).

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

%C Also called the 54-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 54 kinds of squares available. (End)

%H Vincenzo Librandi, <a href="/A042405/b042405.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 (54,1).

%F a(n) = F(n, 54), the n-th Fibonacci polynomial evaluated at x=54. - _T. D. Noe_, Jan 19 2006

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

%F a(n) = 54*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=54.

%F G.f.: 1/(1 - 54*x - x^2). (End)

%t Denominator[Convergents[Sqrt[730], 30]] (* _Vincenzo Librandi_, Jan 21 2014 *)

%Y Cf. A042404.

%Y Row n=54 of A073133, A172236 and A352361 and column k=54 of A157103.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_

%E a(15) from _Colin Barker_, Dec 11 2013

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Last modified March 29 03:41 EDT 2024. Contains 371264 sequences. (Running on oeis4.)