A140455 13-Fibonacci sequence.
0, 1, 13, 170, 2223, 29069, 380120, 4970629, 64998297, 849948490, 11114328667, 145336221161, 1900485203760, 24851643870041, 324971855514293, 4249485765555850, 55568286807740343, 726637214266180309
Offset: 0
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.
- Index entries for linear recurrences with constant coefficients, signature (13,1).
Programs
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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)
Comments