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

0,2

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).

Formula

a(n) = F(n, 16), the n-th Fibonacci polynomial evaluated at x=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

Mathematica

a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*16, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
Denominator[Convergents[Sqrt[65], 30]] (* Vincenzo Librandi, Dec 11 2013 *)

Prog

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

Crossrefs

Cf. A010517, A020822, A040056, A041112.

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

Sequence in context: A223433 A223593 A067223 * A363689 A041482 A320362

Adjacent sequences: A041110 A041111 A041112 * A041114 A041115 A041116

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Status

approved