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

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

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).

Crossrefs

Cf. A041840, A040420.

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

Programs

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 *)

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)

Extensions

Additional term from Colin Barker, Nov 25 2013