A041313 Denominators of continued fraction convergents to sqrt(170).

1, 26, 677, 17628, 459005, 11951758, 311204713, 8103274296, 210996336409, 5494008020930, 143055204880589, 3724929334916244, 96991217912702933, 2525496595065192502, 65759902689607707985, 1712282966524865600112, 44585117032336113310897

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

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

Also called the 26-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 26 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 (26,1).

Crossrefs

Cf. A041312, A040156.

Row n=26 of A073133, A172236 and A352361 and column k=26 of A157103.

Programs

Mathematica

Denominator[Convergents[Sqrt[170], 30]] (* Vincenzo Librandi, Dec 15 2013 *)
LinearRecurrence[{26,1},{1,26},20] (* Harvey P. Dale, Jul 26 2017 *)

Formula

a(n) = F(n, 26), the n-th Fibonacci polynomial evaluated at x=26. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 21 2008: (Start)

a(n) = 26*a(n-1) + a(n-2) for n > 1, a(0)=1, a(1)=26.

G.f.: 1/(1-26*x-x^2). (End)

Extensions

An additional term from Colin Barker, Nov 15 2013