A041545 Denominators of continued fraction convergents to sqrt(290).

1, 34, 1157, 39372, 1339805, 45592742, 1551493033, 52796355864, 1796627592409, 61138134497770, 2080493200516589, 70797906952061796, 2409209329570617653, 81983915112353061998, 2789862323149574725585, 94937302902197893731888

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Michael A. Allen, Jul 13 2023: (Start)

Also called the 34-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 34 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 (34,1).

Crossrefs

Cf. A041544, A040272.

Row n=34 of A073133, A172236 and A352361 and column k=34 of A157103.

Programs

Mathematica

a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*34,{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
Denominator[Convergents[Sqrt[290], 30]] (* Vincenzo Librandi, Dec 20 2013 *)
LinearRecurrence[{34,1},{1,34},20] (* Harvey P. Dale, Oct 08 2021 *)

Formula

a(n) = F(n, 34), the n-th Fibonacci polynomial evaluated at x=34. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 34*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=34.

G.f.: 1/(1-34*x-x^2). (End)

Extensions

More terms from Colin Barker, Nov 18 2013