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A140455 13-Fibonacci sequence. 12
0, 1, 13, 170, 2223, 29069, 380120, 4970629, 64998297, 849948490, 11114328667, 145336221161, 1900485203760, 24851643870041, 324971855514293, 4249485765555850, 55568286807740343, 726637214266180309 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Sergio Falcon, 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).

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(a(n+k)/a(k), k=infinity) = (A088316(n) + A140455(n)*sqrt(173))/2.

Limit(A088316(n)/ A140455(n), n=infinity) = sqrt(173). (End)

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 *)

PROG

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

CROSSREFS

Sequence in context: A057684 A053153 A167254 * A041314 A275293 A296585

Adjacent sequences:  A140452 A140453 A140454 * A140456 A140457 A140458

KEYWORD

nonn,easy

AUTHOR

R. J. Mathar, Jul 22 2008

STATUS

approved

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Last modified December 1 06:09 EST 2020. Contains 338833 sequences. (Running on oeis4.)