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

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

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)

a(2*n) and b(2*n) = A041612(2*n) give all (positive integer) solutions to the Pell equation b^2 - 13*a^2 = -1. a(2*n+1) and b(2*n+1) = A041612(2*n+1) give all (positive integer) solutions to the Pell equation b^2 - 13*a^2 = 1. - Robert FERREOL, Oct 09 2024

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).

Crossrefs

Cf. A041612 (numerators), A040306 (continued fraction), A295330.

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

Programs

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 *)

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)

a(n) = ((18 + 5*sqrt(13))^(n+1) - (18 - 5*sqrt(13))^(n+1)) / (2*sqrt(13)). - Robert FERREOL, Oct 09 2024

Extensions

More terms from Colin Barker, Nov 20 2013