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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

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OFFSET

0,2

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).

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)

Sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = sqrt(122)-11. - Amiram Eldar, Apr 05 2026

MATHEMATICA