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

0,2

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).

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)

Mathematica

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

Crossrefs

Cf. A041684, A040342.

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

Sequence in context: A239364 A078987 A009982 * A221385 A158766 A217224

Adjacent sequences: A041682 A041683 A041684 * A041686 A041687 A041688

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

More terms from Colin Barker, Nov 21 2013

Status

approved