A041481 Denominators of continued fraction convergents to sqrt(257).

1, 32, 1025, 32832, 1051649, 33685600, 1078990849, 34561392768, 1107043559425, 35459955294368, 1135825612979201, 36381879570628800, 1165355971873100801, 37327772979509854432, 1195654091316188442625, 38298258695097540018432, 1226739932334437469032449

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

From Michael A. Allen, May 16 2023: (Start)

Also called the 32-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 32 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 (32,1).

Crossrefs

Cf. A041480, A040240.

Row n=32 of A073133, A172236 and A352361 and column k=32 of A157103.

Programs

Mathematica

a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst,a]; s+=a*32,{n,3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
Denominator[Convergents[Sqrt[257], 30]] (* Vincenzo Librandi, Dec 18 2013 *)
LinearRecurrence[{32,1},{1,32},30] (* Harvey P. Dale, Nov 03 2015 *)

Formula

a(n) = F(n, 32), the n-th Fibonacci polynomial evaluated at x=32. - T. D. Noe, Jan 19 2006

a(n) = 32*a(n-1)+a(n-2) for n>1; a(0)=1, a(1)=32. G.f.: 1/(1-32*x-x^2). [Philippe Deléham, Nov 23 2008]

Extensions

More terms from Colin Barker, Nov 18 2013