A042201 Denominators of continued fraction convergents to sqrt(626).

1, 50, 2501, 125100, 6257501, 313000150, 15656265001, 783126250200, 39171968775001, 1959381565000250, 98008250218787501, 4902371892504375300, 245216602875437552501, 12265732515664382000350, 613531842386094537570001, 30688857851820391260500400

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Michael A. Allen, Dec 02 2023: (Start)

Also called the 50-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 50 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 (50,1).

Crossrefs

Cf. A042200, A040600.

Row n=50 of A073133, A172236 and A352361 and column k=50 of A157103.

Programs

Mathematica

Denominator[Convergents[Sqrt[626], 30]] (* Vincenzo Librandi, Jan 16 2014 *)

Formula

a(n) = F(n, 50), the n-th Fibonacci polynomial evaluated at x=50. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 50*a(n-1) + a(n-2) for n > 1, a(0)=1, a(1)=50.

G.f.: 1/(1 - 50*x - x^2). (End)

Extensions

Additional term from Colin Barker, Dec 04 2013