A041221 Denominators of continued fraction convergents to sqrt(122).

1, 22, 485, 10692, 235709, 5196290, 114554089, 2525386248, 55673051545, 1227332520238, 27056988496781, 596481079449420, 13149640736384021, 289888577279897882, 6390698340894137425, 140885252076950921232, 3105866244033814404529, 68469942620820867820870

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy
• less

Comments

From Michael A. Allen, May 04 2023: (Start)

Also called the 22-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.

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)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (22,1).

Crossrefs

Cf. A040110, A041220.

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

Programs

Magma

[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

Mathematica

Denominator[Convergents[Sqrt[122], 30]]  (* Vincenzo Librandi, Dec 13 2013 *)
Fibonacci[1+Range[0,30], 22] (* G. C. Greubel, Oct 25 2024 *)

SageMath

A041221=BinaryRecurrenceSequence(22,1,1,22)
[A041221(n) for n in range(31)] # G. C. Greubel, Oct 25 2024

Formula

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

From Philippe Deléham, Nov 21 2008: (Start)

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

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

Extensions

More terms from Colin Barker, Nov 14 2013