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%I #36 Apr 30 2023 02:16:41
%S 0,1,13,170,2223,29069,380120,4970629,64998297,849948490,11114328667,
%T 145336221161,1900485203760,24851643870041,324971855514293,
%U 4249485765555850,55568286807740343,726637214266180309
%N 13-Fibonacci sequence.
%C 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.
%C For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - _Johannes W. Meijer_, Jun 12 2010
%C 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
%C 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
%C From _Michael A. Allen_, Apr 21 2023: (Start)
%C Also called the 13-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C 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)
%H Vincenzo Librandi, <a href="/A140455/b140455.txt">Table of n, a(n) for n = 0..900</a>
%H Michael A. Allen and Kenneth Edwards, <a href="https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.
%H Sergio Falcon and Angel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2006.10.022">The k-Fibonacci sequence and Pascal 2-triangle</a>, Chaos, Solit. Fract. 33 (2007) 38-49.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive sequences.</a> [From Johannes W. Meijer, Jun 12 2010]
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,1).
%F O.g.f.: x/(1-13*x-x^2).
%F a(n) = 13*a(n-1) + a(n-2).
%F a(n-r)*a(n+r) - a(n)^2 = (-1)^(n+1-r)*a(r)^2.
%F 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).
%F a(n) = ((13+sqrt(173))^n - (13-sqrt(173))^n)/(2^n*sqrt(173)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009
%F From _Johannes W. Meijer_, Jun 12 2010: (Start)
%F a(2*n) = 13*A097844(n), a(2*n+1) = A098244(n).
%F a(3*n+1) = A041319(5*n), a(3*n+2) = A041319(5*n+3), a(3*n+3) = 2*A041319(5*n+4).
%F Limit_{k->oo} a(n+k)/a(k) = (A088316(n) + A140455(n)*sqrt(173))/2.
%F Limit_{n->oo} A088316(n)/A140455(n) = sqrt(173). (End)
%p 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:
%t LinearRecurrence[{13, 1}, {0, 1}, 30] (* _Vincenzo Librandi_, Nov 17 2012 *)
%o (Sage) [lucas_number1(n,13,-1) for n in range(0, 18)] # _Zerinvary Lajos_, Apr 29 2009
%Y Row n=13 of A073133, A172236 and A352361 and column k=13 of A157103.
%K nonn,easy
%O 0,3
%A _R. J. Mathar_, Jul 22 2008
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