A041925 Denominators of continued fraction convergents to sqrt(485).

1, 44, 1937, 85272, 3753905, 165257092, 7275065953, 320268159024, 14099074063009, 620679526931420, 27323998259045489, 1202876602924932936, 52953894526956094673, 2331174235788993098548, 102624620269242652430785, 4517814466082465700053088

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

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

Also called the 44-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 44 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 (44,1).

Crossrefs

Cf. A041924, A040462.

Row n=44 of A073133, A172236 and A352361 and column k=44 of A157103.

Programs

Mathematica

a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*44,{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
Denominator[Convergents[Sqrt[485],20]] (* Harvey P. Dale, Oct 20 2011 *)
CoefficientList[Series[1/(1 - 44 x - x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 27 2013 *)

Formula

a(n) = F(n, 44), the n-th Fibonacci polynomial evaluated at x=44. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 44*a(n-1) + a(n-2) for n>1; a(0)=1, a(1)=44.

G.f.: 1/(1-44*x-x^2). (End)

Extensions

Additional term from Colin Barker, Nov 27 2013