A041181 Denominators of continued fraction convergents to sqrt(101).

1, 20, 401, 8040, 161201, 3232060, 64802401, 1299280080, 26050404001, 522307360100, 10472197606001, 209966259480120, 4209797387208401, 84405914003648140, 1692328077460171201, 33930967463207072160, 680311677341601614401

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy
• less

Comments

Generalized Pell numbers (A000129). Antidiagonals of A038207. - Mark Dols, Aug 31 2009

a(n) equals the number of words of length n on alphabet {0,1,...,20} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015

From Michael A. Allen, May 03 2023: (Start)

Also called the 20-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 20 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 (20,1).

Crossrefs

Cf. A000129, A038207, A040090, A041180.

Cf. similar sequences listed in A243399.

Row n=20 of A073133, A172236 and A352361 and column k=20 of A157103.

Programs

Magma

[n le 2 select (20)^(n-1) else 20*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013

Mathematica

Denominator[Convergents[Sqrt[101], 30]] (* Vincenzo Librandi, Dec 12 2013 *)
LinearRecurrence[{20,1},{1,20},20] (* Harvey P. Dale, Mar 17 2020 *)

SageMath

A041181=BinaryRecurrenceSequence(20,1,1,20)
[A041181(n) for n in range(31)] # G. C. Greubel, Sep 29 2024

Formula

a(n) = Fibonacci(n+1, 20), the n-th Fibonacci polynomial evaluated at x=20. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 21 2008: (Start)

a(n) = 20*a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=20.

G.f.: 1/(1-20*x-x^2). (End)