OFFSET
0,2
COMMENTS
From Michael A. Allen, Jul 13 2023: (Start)
Also called the 36-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 36 kinds of squares available. (End)
a(2*n) and b(2*n) = A041612(2*n) give all (positive integer) solutions to the Pell equation b^2 - 13*a^2 = -1. a(2*n+1) and b(2*n+1) = A041612(2*n+1) give all (positive integer) solutions to the Pell equation b^2 - 13*a^2 = 1. - Robert FERREOL, Oct 09 2024
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 (36,1).
FORMULA
a(n) = F(n, 36), the n-th Fibonacci polynomial evaluated at x=36. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 36*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=36.
G.f.: 1/(1 - 36*x - x^2). (End)
a(n) = ((18 + 5*sqrt(13))^(n+1) - (18 - 5*sqrt(13))^(n+1)) / (2*sqrt(13)). - Robert FERREOL, Oct 09 2024
MAPLE
with (combinat):seq(fibonacci(3*n, 3)/10, n=1..15); # Zerinvary Lajos, Apr 20 2008
MATHEMATICA
a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*36, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
Denominator[Convergents[Sqrt[325], 30]] (* Vincenzo Librandi Dec 21 2013 *)
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
EXTENSIONS
More terms from Colin Barker, Nov 20 2013
STATUS
approved