A041685 Denominators of continued fraction convergents to sqrt(362).

1, 38, 1445, 54948, 2089469, 79454770, 3021370729, 114891542472, 4368899984665, 166133090959742, 6317426356454861, 240228334636244460, 9134994142533744341, 347370005750918529418, 13209195212677437862225, 502296788087493557293968

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

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

Also called the 38-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 38 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 (38,1).

Crossrefs

Cf. A041684, A040342.

Row n=38 of A073133, A172236 and A352361 and column k=38 of A157103.

Programs

Mathematica

Denominator[Convergents[Sqrt[362], 30]] (* Vincenzo Librandi, Dec 22 2013 *)
LinearRecurrence[{38,1},{1,38},30] (* Harvey P. Dale, May 23 2017 *)

Formula

a(n) = F(n, 38), the n-th Fibonacci polynomial evaluated at x=38. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 38*a(n-1) + a(n-2), n > 1; a(0)=1, a(1)=38.

G.f.: 1/(1-38*x-x^2). (End)

Extensions

More terms from Colin Barker, Nov 21 2013