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

0,2

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).

Formula

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

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). - Philippe Deléham, Nov 21 2008

Mathematica

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

Prog

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

Crossrefs

Cf. A041180, A040090.

Cf. similar sequences listed in A243399.

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

Sequence in context: A007545 A055476 A223180 * A041762 A196740 A196898

Adjacent sequences: A041178 A041179 A041180 * A041182 A041183 A041184

Keyword

nonn,frac,easy,less

Author

N. J. A. Sloane

Status

approved