|
%I #56 Apr 30 2023 11:47:09
%S 1,14,197,2772,39005,548842,7722793,108667944,1529074009,21515704070,
%T 302748930989,4260000737916,59942759261813,843458630403298,
%U 11868363584907985,167000548819115088,2349876047052519217,33065265207554384126,465263588952813896981
%N Denominators of continued fraction convergents to sqrt(50).
%C 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
%C 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
%C From _Michael A. Allen_, Apr 30 2023: (Start)
%C Also called the 14-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C 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)
%H Vincenzo Librandi, <a href="/A041085/b041085.txt">Table of n, a(n) for n = 0..800</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 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,1).
%F a(n) = round((7+5*sqrt(2))*a(n-1)). - _Vladeta Jovovic_, Jun 15 2003
%F From _Paul Barry_, Feb 06 2004: (Start)
%F a(n) = A000129(3n+3)/5.
%F a(n) = (1+sqrt(2))^(3*n)*(1/2+7*sqrt(2)/20)+(1-sqrt(2))^(3*n)*(1/2-7*sqrt(2)/20).
%F a(n) = Sum_{i=0..n} Sum_{j=0..n} (n!/(i!j!(n-i-j)!)*A000129(2n-i)/5. (End)
%F a(n) = F(n, 14), the n-th Fibonacci polynomial evaluated at x=14. - _T. D. Noe_, Jan 19 2006
%F From _Philippe Deléham_, Nov 03 2008: (Start)
%F a(n) = 14*a(n-1) + a(n-2); a(0)=1, a(1)=14.
%F G.f.: 1/(1-14*x-x^2). (End)
%F a(n) = ((7+5*sqrt(2))^(n+1) - (7-5*sqrt(2))^(n+1))/(10*sqrt(2)). - _Gerry Martens_, Jul 11 2015
%p with(combinat): seq(fibonacci(3*n+3,2)/5, n=0..17); # _Zerinvary Lajos_, Apr 20 2008
%t LinearRecurrence[{14, 1}, {1, 14}, 30] (* _Vincenzo Librandi_, Nov 17 2012 *)
%t Table[Fibonacci[3n + 3, 2]/5, {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 16 2016 *)
%o (Magma) I:=[1, 14]; [n le 2 select I[n] else 14*Self(n-1) +Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 17 2012
%Y Cf. A041084, A040042, A020807.
%Y Row n=14 of A073133, A172236 and A352361 and column k=14 of A157103.
%K nonn,cofr,easy,frac
%O 0,2
%A _N. J. A. Sloane_
%E Additional term from _Colin Barker_, Nov 12 2013
| 