A041113 Denominators of continued fraction convergents to sqrt(65).

1, 16, 257, 4128, 66305, 1065008, 17106433, 274767936, 4413393409, 70889062480, 1138638393089, 18289103351904, 293764292023553, 4718517775728752, 75790048703683585, 1217359297034666112, 19553538801258341377, 314073980117168128144, 5044737220675948391681, 81029869510932342395040

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

Sqrt(65) = 16/2 + 16/257 + 16/(257*66305) + 16/(66305*17106433) + ... - Gary W. Adamson, Jun 13 2008

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 16'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,...,16} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015

From Michael A. Allen, May 01 2023: (Start)

Also called the 16-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 16 kinds of squares available. (End)

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..500
• 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 (16,1).

Crossrefs

Cf. A010517, A020822, A040056, A041112.

Row n=16 of A073133, A172236 and A352361 and column k=16 of A157103.

Programs

Magma

[n le 2 select (16)^(n-1) else 16*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

Mathematica

Denominator[Convergents[Sqrt[65], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
Fibonacci[Range[30], 16] (* G. C. Greubel, Sep 29 2024 *)

SageMath

A041113=BinaryRecurrenceSequence(16,1,1,16)
[A041113(n) for n in range(0,31)] # G. C. Greubel, Sep 29 2024

Formula

a(n) = Fibonacci(n+1, 16). - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 21 2008: (Start)

a(n) = 16*a(n-1) + a(n-2) for n > 1; a(0) = 1, a(1) = 16.

G.f.: 1/(1 - 16*x - x^2). (End)

a(n) = ((8+sqrt(65))^(n+1) - (8-sqrt(65))^(n+1))/(2*sqrt(65)). - Rolf Pleisch, May 14 2011

E.g.f.: exp(8*x)*(cosh(sqrt(65)*x) + 8*sinh(sqrt(65)*x)/sqrt(65)). - Stefano Spezia, Oct 28 2022