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A041365 Denominators of continued fraction convergents to sqrt(197). 3

%I #37 Jul 03 2023 20:31:46

%S 1,28,785,22008,617009,17298260,484968289,13596410352,381184458145,

%T 10686761238412,299610499133681,8399780736981480,235493471134615121,

%U 6602216972506204868,185097568701308351425,5189334140609140044768

%N Denominators of continued fraction convergents to sqrt(197).

%C From _Michael A. Allen_, May 16 2023: (Start)

%C Also called the 28-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 28 kinds of squares available. (End)

%H Vincenzo Librandi, <a href="/A041365/b041365.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 (28,1).

%F a(n) = F(n, 28), the n-th Fibonacci polynomial evaluated at x=28. - _T. D. Noe_, Jan 19 2006

%F From _Philippe Deléham_, Nov 21 2008: (Start)

%F a(n) = 28*a(n-1) + a(n-2), n > 1; a(0)= 1, a(1)=28.

%F G.f.: 1/(1-28*x-x^2). (End)

%t a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*28,{n,3*4!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Oct 27 2009 *)

%t Denominator[Convergents[Sqrt[197], 30]] (* _Vincenzo Librandi_, Dec 16 2013 *)

%t LinearRecurrence[{28,1},{1,28},20] (* _Harvey P. Dale_, Mar 07 2021 *)

%Y Cf. A041364, A040182.

%Y Row n=28 of A073133, A172236 and A352361 and column k=28 of A157103.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_

%E Additional term from _Colin Barker_, Nov 16 2013

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)