OFFSET
0,2
COMMENTS
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 14's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
a(n) equals the number of words of length n on alphabet {0,1,...,14} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Apr 30 2023: (Start)
Also called the 14-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 14 kinds of squares available. (End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..800
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 (14,1).
FORMULA
a(n) = round((7+5*sqrt(2))*a(n-1)). - Vladeta Jovovic, Jun 15 2003
From Paul Barry, Feb 06 2004: (Start)
a(n) = A000129(3*n+3)/5.
a(n) = (1/20)*((10+7*sqrt(2))*(1+sqrt(2))^(3*n) + (10-7*sqrt(2))*(1-sqrt(2))^(3*n)).
a(n-1) = Sum_{i=0..n} Sum_{j=0..n-i} (n!/(i!*j!*(n-i-j)!))*A000129(2*n-i)/5. (End)
a(n) = Fibonacci(n+1, 14), the n-th Fibonacci polynomial evaluated at x=14. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 03 2008: (Start)
a(n) = 14*a(n-1) + a(n-2); a(0)=1, a(1)=14.
G.f.: 1/(1-14*x-x^2). (End)
a(n) = ((7+5*sqrt(2))^(n+1) - (7-5*sqrt(2))^(n+1))/(10*sqrt(2)). - Gerry Martens, Jul 11 2015
Sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = 5*sqrt(2)-7. - Amiram Eldar, Apr 05 2026
E.g.f.: exp(7*x)*(10*cosh(5*sqrt(2)*x) + 7*sqrt(2)*sinh(5*sqrt(2)*x))/10. - Stefano Spezia, Apr 05 2026
MAPLE
with(combinat): seq(fibonacci(3*n+3, 2)/5, n=0..17); # Zerinvary Lajos, Apr 20 2008
MATHEMATICA
LinearRecurrence[{14, 1}, {1, 14}, 30] (* Vincenzo Librandi, Nov 17 2012 *)
Table[Fibonacci[3n + 3, 2]/5, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
Convergents[Sqrt[50], 20]//Denominator (* Harvey P. Dale, Aug 16 2025 *)
PROG
(Magma) [n le 2 select (14)^(n-1) else 14*Self(n-1) +Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 17 2012
(SageMath)
A041085=BinaryRecurrenceSequence(14, 1, 1, 14)
[A041085(n) for n in range(31)] # G. C. Greubel, Sep 29 2024
CROSSREFS
KEYWORD
nonn,cofr,easy,frac,changed
AUTHOR
EXTENSIONS
Additional term from Colin Barker, Nov 12 2013
STATUS
approved