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 A041085 Denominators of continued fraction convergents to sqrt(50). 15
 1, 14, 197, 2772, 39005, 548842, 7722793, 108667944, 1529074009, 21515704070, 302748930989, 4260000737916, 59942759261813, 843458630403298, 11868363584907985, 167000548819115088, 2349876047052519217, 33065265207554384126, 465263588952813896981 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 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 From Michael A. Allen, Apr 30 2023: (Start) Also called the 14-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence. 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) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..800 Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17. Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (14,1). FORMULA a(n) = round((7+5*sqrt(2))*a(n-1)). - Vladeta Jovovic, Jun 15 2003 From Paul Barry, Feb 06 2004: (Start) a(n) = A000129(3n+3)/5. a(n) = (1+sqrt(2))^(3*n)*(1/2+7*sqrt(2)/20)+(1-sqrt(2))^(3*n)*(1/2-7*sqrt(2)/20). a(n) = Sum_{i=0..n} Sum_{j=0..n} (n!/(i!j!(n-i-j)!)*A000129(2n-i)/5. (End) a(n) = F(n, 14), the n-th Fibonacci polynomial evaluated at x=14. - T. D. Noe, Jan 19 2006 From Philippe Deléham, Nov 03 2008: (Start) a(n) = 14*a(n-1) + a(n-2); a(0)=1, a(1)=14. G.f.: 1/(1-14*x-x^2). (End) a(n) = ((7+5*sqrt(2))^(n+1) - (7-5*sqrt(2))^(n+1))/(10*sqrt(2)). - Gerry Martens, Jul 11 2015 MAPLE with(combinat): seq(fibonacci(3*n+3, 2)/5, n=0..17); # Zerinvary Lajos, Apr 20 2008 MATHEMATICA LinearRecurrence[{14, 1}, {1, 14}, 30] (* Vincenzo Librandi, Nov 17 2012 *) Table[Fibonacci[3n + 3, 2]/5, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *) PROG (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 CROSSREFS Cf. A041084, A040042, A020807. Row n=14 of A073133, A172236 and A352361 and column k=14 of A157103. Sequence in context: A278476 A067221 A072533 * A124239 A041366 A051817 Adjacent sequences: A041082 A041083 A041084 * A041086 A041087 A041088 KEYWORD nonn,cofr,easy,frac AUTHOR N. J. A. Sloane EXTENSIONS Additional term from Colin Barker, Nov 12 2013 STATUS approved

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Last modified September 10 21:37 EDT 2024. Contains 375795 sequences. (Running on oeis4.)