A041085 - OEIS

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

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

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)

FORMULA

a(n) = round((7+5*sqrt(2))*a(n-1)). - Vladeta Jovovic, Jun 15 2003

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

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 *)

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)