A041421 - OEIS

%I #40 Jan 05 2025 19:51:35

%S 1,30,901,27060,812701,24408090,733055401,22016070120,661215159001,

%T 19858470840150,596415340363501,17912318681745180,537965975792718901,

%U 16156891592463312210,485244713749692085201,14573498304083225868240

%N Denominators of continued fraction convergents to sqrt(226).

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

%C Also called the 30-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 30 kinds of squares available. (End)

%H Vincenzo Librandi, <a href="/A041421/b041421.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 (30,1).

%F a(n) = F(n, 30), the n-th Fibonacci polynomial evaluated at x=30. - _T. D. Noe_, Jan 19 2006

%F From _Philippe Deléham_, Nov 22 2008: (Start)

%F a(n) = 30*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=30.

%F G.f.: 1/(1-30*x-x^2). (End)

%t Denominator[Convergents[Sqrt[226], 30]] (* _Vincenzo Librandi_, Dec 17 2013 *)

%t LinearRecurrence[{30,1},{1,30},20] (* _Harvey P. Dale_, Jun 30 2022 *)

%Y Cf. A041420, A040210.

%Y Row n=30 of A073133, A172236 and A352361 and column k=30 of A157103.

%K nonn,frac,easy,changed

%O 0,2

%A _N. J. A. Sloane_

%E Additional term from _Colin Barker_, Nov 17 2013