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

0,2

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).

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)

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 *)

Crossrefs

Cf. A041544, A040272.

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

Sequence in context: A264134 A264019 A009978 * A189434 A167258 A251924

Adjacent sequences: A041542 A041543 A041544 * A041546 A041547 A041548

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

More terms from Colin Barker, Nov 18 2013

Status

approved