A041613 Denominators of continued fraction convergents to sqrt(325).

1, 36, 1297, 46728, 1683505, 60652908, 2185188193, 78727427856, 2836372591009, 102188140704180, 3681609437941489, 132640127906597784, 4778726214075461713, 172166783834623219452, 6202782944260511361985, 223472352777213032250912

Offset

0,2

Comments

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

Also called the 36-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 36 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 (36,1).

Formula

a(n) = F(n, 36), the n-th Fibonacci polynomial evaluated at x=36. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 36*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=36.

G.f.: 1/(1 - 36*x - x^2). (End)

Maple

with (combinat):seq(fibonacci(3*n, 3)/10, n=1..15); # Zerinvary Lajos, Apr 20 2008

Mathematica

a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*36, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
Denominator[Convergents[Sqrt[325], 30]] (* Vincenzo Librandi Dec 21 2013 *)

Crossrefs

Cf. A041612, A040306.

Row n=36 of A073133, A172236 and A352361 and column k=36 of A157103.

Sequence in context: A224011 A300357 A009980 * A255821 A209042 A283729

Adjacent sequences: A041610 A041611 A041612 * A041614 A041615 A041616

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

More terms from Colin Barker, Nov 20 2013

Status

approved