A041085 Denominators of continued fraction convergents to sqrt(50).

1, 14, 197, 2772, 39005, 548842, 7722793, 108667944, 1529074009, 21515704070, 302748930989, 4260000737916, 59942759261813, 843458630403298, 11868363584907985, 167000548819115088, 2349876047052519217, 33065265207554384126, 465263588952813896981

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(3n+3)/5.

a(n) = (1+sqrt(2))^(3*n)*(1/2+7*sqrt(2)/20)+(1-sqrt(2))^(3*n)*(1/2-7*sqrt(2)/20).

a(n) = Sum_{i=0..n} Sum_{j=0..n} (n!/(i!j!(n-i-j)!)*A000129(2n-i)/5. (End)

a(n) = F(n, 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

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 *)

Prog

(Magma) I:=[1, 14]; [n le 2 select I[n] else 14*Self(n-1) +Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 17 2012

Crossrefs

Cf. A041084, A040042, A020807.

Row n=14 of A073133, A172236 and A352361 and column k=14 of A157103.

Sequence in context: A278476 A067221 A072533 * A124239 A041366 A051817

Adjacent sequences: A041082 A041083 A041084 * A041086 A041087 A041088

Keyword

nonn,cofr,easy,frac

Author

N. J. A. Sloane

Extensions

Additional term from Colin Barker, Nov 12 2013

Status

approved