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A041841 Denominators of continued fraction convergents to sqrt(442). 3

%I #30 Sep 03 2023 10:24:14

%S 1,42,1765,74172,3116989,130987710,5504600809,231324221688,

%T 9721121911705,408518444513298,17167495791470221,721443341686262580,

%U 30317787846614498581,1274068532899495202982,53541196169625413023825,2250004307657166842203632

%N Denominators of continued fraction convergents to sqrt(442).

%C From _Michael A. Allen_, Aug 08 2023: (Start)

%C Also called the 42-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 42 kinds of squares available. (End)

%H Vincenzo Librandi, <a href="/A041841/b041841.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 (42,1).

%F a(n) = F(n, 42), the n-th Fibonacci polynomial evaluated at x=42. - _T. D. Noe_, Jan 19 2006

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

%F a(n) = 42*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=42.

%F G.f.: 1/(1 - 42*x - x^2).

%F (End)

%t a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*42,{n,3*4!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)

%t Denominator[Convergents[Sqrt[442], 30]] (* _Vincenzo Librandi_, Dec 25 2013 *)

%Y Cf. A041840, A040420.

%Y Row n=42 of A073133, A172236 and A352361 and column k=42 of A157103.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_

%E Additional term from _Colin Barker_, Nov 25 2013

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)