A041221 - OEIS

%I #43 Jan 05 2025 19:51:35

%S 1,22,485,10692,235709,5196290,114554089,2525386248,55673051545,

%T 1227332520238,27056988496781,596481079449420,13149640736384021,

%U 289888577279897882,6390698340894137425,140885252076950921232,3105866244033814404529,68469942620820867820870

%N Denominators of continued fraction convergents to sqrt(122).

%C From _Michael A. Allen_, May 04 2023: (Start)

%C Also called the 22-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 22 kinds of squares available. (End)

%H Vincenzo Librandi, <a href="/A041221/b041221.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 (22,1).

%F a(n) = F(n, 22), the n-th Fibonacci polynomial evaluated at x=22. - _T. D. Noe_, Jan 19 2006

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

%F a(n) = 22*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=22.

%F G.f.: 1/(1 - 22*x - x^2). (End)

%t Denominator[Convergents[Sqrt[122], 30]]  (* _Vincenzo Librandi_, Dec 13 2013 *)

%t Fibonacci[1+Range[0,30], 22] (* _G. C. Greubel_, Oct 25 2024 *)

%o (Magma)

%o [n le 2 select (22)^(n-1) else 22*Self(n-1)+Self(n-2): n in [1..31]]; // _G. C. Greubel_, Oct 25 2024

%o (SageMath)

%o A041221=BinaryRecurrenceSequence(22,1,1,22)

%o [A041221(n) for n in range(31)] # _G. C. Greubel_, Oct 25 2024

%Y Cf. A040110, A041220.

%Y Row n=22 of A073133, A172236 and A352361 and column k=22 of A157103.

%K nonn,frac,easy,less

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Colin Barker_, Nov 14 2013