OFFSET
0,2
COMMENTS
a(n) equals the number of words of length n on alphabet {0,1,...,20} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, May 03 2023: (Start)
Also called the 20-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 20 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 (20,1).
FORMULA
a(n) = Fibonacci(n+1, 20), the n-th Fibonacci polynomial evaluated at x=20. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 20*a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=20.
G.f.: 1/(1-20*x-x^2). (End)
MATHEMATICA
Denominator[Convergents[Sqrt[101], 30]] (* Vincenzo Librandi, Dec 12 2013 *)
LinearRecurrence[{20, 1}, {1, 20}, 20] (* Harvey P. Dale, Mar 17 2020 *)
PROG
(Magma) [n le 2 select (20)^(n-1) else 20*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013
(SageMath)
A041181=BinaryRecurrenceSequence(20, 1, 1, 20)
[A041181(n) for n in range(31)] # G. C. Greubel, Sep 29 2024
CROSSREFS
KEYWORD
nonn,frac,easy,less
AUTHOR
STATUS
approved