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 A005592 a(n) = F(2n+1) + F(2n-1) - 1. (Formerly M1619) 8
 1, 2, 6, 17, 46, 122, 321, 842, 2206, 5777, 15126, 39602, 103681, 271442, 710646, 1860497, 4870846, 12752042, 33385281, 87403802, 228826126, 599074577, 1568397606, 4106118242, 10749957121, 28143753122, 73681302246, 192900153617, 505019158606, 1322157322202 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For any m, the maximum element in the continued fraction of F(2n+m)/F(m) is a(n). - Benoit Cloitre, Jan 10 2006 The continued fraction [a(n);1,a(n)-1,1,a(n)-1,...] = phi^(2n), where phi = 1.618... is the golden ratio, A001622. - Thomas Ordowski, Jun 07 2013 a(n) is the number of labeled subgraphs of the n-cycle C_n. For example, a(3)=17. There are 7 subgraphs of the triangle C_3 with 0 edges, 6 with 1 edge, 3 with 2 edges, and 1 with 3 edges (C_3 itself); here 7+6+3+1 = 17. - John P. McSorley, Oct 31 2016 a(n) equals the row sums of triangle A277919. - John P. McSorley, Nov 25 2016 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..200 from Vincenzo Librandi) M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392. M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392. (Annotated scanned copy) Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. J. Salas and A. D. Sokal, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial, J. Stat. Phys. 135 (2009) 279-373, arXiv:0711.1738 [cond-mat.stat-mech]. Mentions this sequence. - N. J. A. Sloane, Mar 14 2014 Index entries for linear recurrences with constant coefficients, signature (4,-4,1). FORMULA a(n) = Lucas(2*n)-1, with Lucas(n)=A000032(n). a(n) = floor(r^(2*n)), where r = golden ratio = (1+sqrt(5))/2. a(n) = floor(Fibonacci(5*n)/Fibonacci(3*n)). - Gary Detlefs, Mar 11 2011 a(n) = +4*a(n-1) -4*a(n-2) +1*a(n-3). - Joerg Arndt, Mar 11 2011 a(n) = A001519(2*n-1) + A001519(2*n+1) - 1. - Reinhard Zumkeller, Aug 09 2013 a(n) = 3*a(n) - a(n-1) + 1; a(n) = A004146(n) + 1, n>0. - Richard R. Forberg, Sep 04 2013 a(n) = 2*cosh(2*n*arcsinh(1/2)) - 1. - Ilya Gutkovskiy, Oct 31 2016 EXAMPLE G.f. = 1 + 2*x + 6*x^2 + 17*x^3 + 46*x^4 + 122*x^5 + 321*x^6 + 842*x^7 + ... MAPLE A005592:=-(2-2*z+z**2)/(z-1)/(z**2-3*z+1); # conjectured by Simon Plouffe in his 1992 dissertation # second Maple program: F:= n-> (<<0|1>, <1|1>>^n)[1, 2]: a:= n-> F(2*n+1)+F(2*n-1)-1: seq(a(n), n=0..30);  # Alois P. Heinz, Nov 04 2016 MATHEMATICA Table[Fibonacci[2n+1]+Fibonacci[2n-1]-1, {n, 30}] (* Harvey P. Dale, Aug 22 2011 *) a[n_] := LucasL[2n]-1; Array[a, 30] (* Jean-François Alcover, Dec 09 2015 *) PROG (Sage) [lucas_number2(n, 3, 1)-1 for n in xrange(1, 29)] # Zerinvary Lajos, Jul 06 2008 (MAGMA) [Fibonacci(2*n+1)+Fibonacci(2*n-1)-1: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011 (PARI) a(n)=fibonacci(2*n+1)+fibonacci(2*n-1)-1 \\ Charles R Greathouse IV, Aug 23 2011 (Haskell) a005592 n = a005592_list !! (n-1) a005592_list = map (subtract 1) \$                    tail \$ zipWith (+) a001519_list \$ tail a001519_list -- Reinhard Zumkeller, Aug 09 2013 CROSSREFS Equals A004146+1 and A005248+1. Bisection of A014217; the other bisection is A002878, which also bisects A000032. Cf. A065034. Sequence in context: A316591 A222115 A190050 * A102403 A278428 A200379 Adjacent sequences:  A005589 A005590 A005591 * A005593 A005594 A005595 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Formulae and comments by Clark Kimberling, Nov 24 2010 a(0)=1 prepended by Alois P. Heinz, Nov 04 2016 STATUS approved

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Last modified February 16 03:43 EST 2019. Contains 320140 sequences. (Running on oeis4.)