A042625 Denominators of continued fraction convergents to sqrt(842).

1, 58, 3365, 195228, 11326589, 657137390, 38125295209, 2211924259512, 128329732346905, 7445336400380002, 431957840954387021, 25061000111754827220, 1453969964322734365781, 84355318930830348042518, 4894062467952482920831825, 283939978460174839756288368

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Michael A. Allen, Jan 22 2024: (Start)

Also called the 58-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 58 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 (58,1).

Crossrefs

Cf. A042624, A040812.

Row n=58 of A073133, A172236 and A352361 and column k=58 of A157103.

Programs

Mathematica

Denominator[Convergents[Sqrt[842], 30]] (* Vincenzo Librandi, Jan 26 2014 *)

Formula

a(n) = F(n, 58), the n-th Fibonacci polynomial evaluated at x=58. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 58*a(n-1) + a(n-2) for n>1; a(0)=1, a(1)=58.

G.f.: 1/(1 - 58*x - x^2). (End)

Extensions

Additional term from Colin Barker, Dec 20 2013