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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• frac

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).

Crossrefs

Cf. A000129, A020807, A040042, A041084.

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

Programs

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

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 *)

SageMath

A041085=BinaryRecurrenceSequence(14,1,1,14)
[A041085(n) for n in range(31)] # G. C. Greubel, Sep 29 2024

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

Extensions

Additional term from Colin Barker, Nov 12 2013