A041841 Denominators of continued fraction convergents to sqrt(442).

1, 42, 1765, 74172, 3116989, 130987710, 5504600809, 231324221688, 9721121911705, 408518444513298, 17167495791470221, 721443341686262580, 30317787846614498581, 1274068532899495202982, 53541196169625413023825, 2250004307657166842203632

Offset

0,2

Comments

From Michael A. Allen, Aug 08 2023: (Start)

Also called the 42-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 42 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 (42,1).

Formula

a(n) = F(n, 42), the n-th Fibonacci polynomial evaluated at x=42. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 42*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=42.

G.f.: 1/(1 - 42*x - x^2).

(End)

Mathematica

a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*42, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
Denominator[Convergents[Sqrt[442], 30]] (* Vincenzo Librandi, Dec 25 2013 *)

Crossrefs

Cf. A041840, A040420.

Row n=42 of A073133, A172236 and A352361 and column k=42 of A157103.

Sequence in context: A207716 A208077 A009986 * A236270 A216703 A258393

Adjacent sequences: A041838 A041839 A041840 * A041842 A041843 A041844

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

Additional term from Colin Barker, Nov 25 2013

Status

approved