A041761 - OEIS

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

%S 1,40,1601,64080,2564801,102656120,4108809601,164455040160,

%T 6582310416001,263456871680200,10544857177624001,422057743976640240,

%U 16892854616243233601,676136242393705984280,27062342550364482604801,1083169838256973010176320

%N Denominators of continued fraction convergents to sqrt(401).

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

%C Also called the 40-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 40 kinds of squares available. (End)

%H Vincenzo Librandi, <a href="/A041761/b041761.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 (40,1).

%F a(n) = F(n, 40), the n-th Fibonacci polynomial evaluated at x=40. - _T. D. Noe_, Jan 19 2006

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

%F a(n) = 40*a(n-1) + a(n-2), n>1; a(0)=1, a(1)=40.

%F G.f.: 1/(1 - 40*x - x^2). (End)

%t a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*40,{n,3*4!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 03 2009 *)

%t Denominator[Convergents[Sqrt[401],20]] (* _Harvey P. Dale_, Aug 18 2012 *)

%t CoefficientList[Series[1/(1 - 40 x - x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 24 2013 *)

%Y Cf. A041760, A040380.

%Y Row n=40 of A073133, A172236 and A352361 and column k=40 of A157103.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_