A140455 13-Fibonacci sequence.

0, 1, 13, 170, 2223, 29069, 380120, 4970629, 64998297, 849948490, 11114328667, 145336221161, 1900485203760, 24851643870041, 324971855514293, 4249485765555850, 55568286807740343, 726637214266180309

Offset: 0

Author

R. J. Mathar, Jul 22 2008

Keywords

• nonn
• easy

Comments

The k-Fibonacci sequences for k=2..12 are A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A041041, A049666, A041061. This here is k=13. k=14 is A041085, k=16 A041113, k=18 A041145, k=20 A041181, k=22 A041221.

For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010

For n>=2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 13's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

For n>=1, a(n) equals the number of words of length n-1 on alphabet {0,1,...,13} avoiding runs of zeros of odd length. - Milan Janjic, Jan 28 2015

From Michael A. Allen, Apr 21 2023: (Start)

Also called the 13-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.

a(n+1) 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 13 kinds of squares available. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..900
• Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
• Sergio Falcon and Angel Plaza, The k-Fibonacci sequence and Pascal 2-triangle, Chaos, Solit. Fract. 33 (2007) 38-49.
• Tanya Khovanova, Recursive sequences. [From Johannes W. Meijer, Jun 12 2010]
• Index entries for linear recurrences with constant coefficients, signature (13,1).

Crossrefs

Row n=13 of A073133, A172236 and A352361 and column k=13 of A157103.

Programs

Maple

F := proc(n,k) coeftayl( x/(1-k*x-x^2),x=0,n) ; end: for n from 0 to 20 do printf("%d,",F(n,13)) ; od:

Mathematica

LinearRecurrence[{13, 1}, {0, 1}, 30] (* Vincenzo Librandi, Nov 17 2012 *)

Sage

[lucas_number1(n,13,-1) for n in range(0, 18)] # Zerinvary Lajos, Apr 29 2009

Formula

O.g.f.: x/(1-13*x-x^2).

a(n) = 13*a(n-1) + a(n-2).

a(n-r)*a(n+r) - a(n)^2 = (-1)^(n+1-r)*a(r)^2.

a(n) = Sum_{i=0..floor((n-1)/2)} binomial(n,2i+1)*13^(n-1-2*i)*(13^2+4)^i/2^(n-1).

a(n) = ((13+sqrt(173))^n - (13-sqrt(173))^n)/(2^n*sqrt(173)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009

From Johannes W. Meijer, Jun 12 2010: (Start)

a(2*n) = 13*A097844(n), a(2*n+1) = A098244(n).

a(3*n+1) = A041319(5*n), a(3*n+2) = A041319(5*n+3), a(3*n+3) = 2*A041319(5*n+4).

Limit_{k->oo} a(n+k)/a(k) = (A088316(n) + A140455(n)*sqrt(173))/2.

Limit_{n->oo} A088316(n)/A140455(n) = sqrt(173). (End)