A041145 Denominators of continued fraction convergents to sqrt(82).

1, 18, 325, 5868, 105949, 1912950, 34539049, 623615832, 11259624025, 203296848282, 3670602893101, 66274148924100, 1196605283526901, 21605169252408318, 390089651826876625, 7043218902136187568, 127168029890278252849, 2296067756927144738850, 41456387654578883552149

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 18's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

a(n) equals the number of words of length n on alphabet {0,1,...,18} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015

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

Also called the 18-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 18 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 (18,1).

Crossrefs

Cf. A010533, A020839, A040072, A041144.

Cf. similar sequences listed in A243399.

Row n=18 of A073133, A172236 and A352361 and column k=18 of A157103.

Programs

Magma

[n le 2 select (18)^(n-1) else 18*Self(n-1)+Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 29 2024

Mathematica

Denominator[Convergents[Sqrt[82], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
Fibonacci[Range[30], 18] (* G. C. Greubel, Sep 29 2024 *)

SageMath

A041145=BinaryRecurrenceSequence(18,1,1,18)
[A041145(n) for n in range(31)] # G. C. Greubel, Sep 29 2024

Formula

a(n) = Fibonacci(n+1, 18), the n-th Fibonacci polynomial evaluated at x=18. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 21 2008: (Start)

a(n) = 18*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=18.

G.f.: 1/(1 - 18*x - x^2). (End)

E.g.f.: exp(9*x)*(cosh(sqrt(82)*x) + 9*sinh(sqrt(82)*x)/sqrt(82)). - Stefano Spezia, Oct 02 2024