A041421 Denominators of continued fraction convergents to sqrt(226).

1, 30, 901, 27060, 812701, 24408090, 733055401, 22016070120, 661215159001, 19858470840150, 596415340363501, 17912318681745180, 537965975792718901, 16156891592463312210, 485244713749692085201, 14573498304083225868240

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Michael A. Allen, May 16 2023: (Start)

Also called the 30-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 30 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 (30,1).

Crossrefs

Cf. A041420, A040210.

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

Programs

Mathematica

Denominator[Convergents[Sqrt[226], 30]] (* Vincenzo Librandi, Dec 17 2013 *)
LinearRecurrence[{30,1},{1,30},20] (* Harvey P. Dale, Jun 30 2022 *)

Formula

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

From Philippe Deléham, Nov 22 2008: (Start)

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

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

Extensions

Additional term from Colin Barker, Nov 17 2013