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%I M1619 #124 Oct 27 2023 20:54:52
%S 1,2,6,17,46,122,321,842,2206,5777,15126,39602,103681,271442,710646,
%T 1860497,4870846,12752042,33385281,87403802,228826126,599074577,
%U 1568397606,4106118242,10749957121,28143753122,73681302246,192900153617,505019158606,1322157322202
%N a(n) = F(2n+1) + F(2n-1) - 1.
%C For any m, the maximum element in the continued fraction of F(2n+m)/F(m) is a(n). - _Benoit Cloitre_, Jan 10 2006
%C 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
%C 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
%C a(n) equals the sum of the n-th row of triangle A277919. - _John P. McSorley_, Nov 25 2016
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A005592/b005592.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 1..200 from Vincenzo Librandi)
%H M. D. McIlroy, <a href="http://dx.doi.org/10.1093/comjnl/25.3.388">The number of states of a dynamic storage system</a>, Computer J., Vol. 25, No. 3 (1982), pp. 388-392.
%H M. D. McIlroy, <a href="/A005207/a005207.pdf">The number of states of a dynamic storage system</a>, Computer J., Vol. 25, No. 3 (1982), pp. 388-392. (Annotated scanned copy)
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Jesús Salas and Alan D. Sokal, <a href="https://doi.org/10.1007/s10955-009-9725-1">Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial</a>, J. Stat. Phys., Vol. 135 (2009), pp. 279-373; <a href="http://arxiv.org/abs/0711.1738">arXiv preprint</a>, arXiv:0711.1738 [cond-mat.stat-mech], 2007-2009. Mentions this sequence. - N. J. A. Sloane, Mar 14 2014
%H Robert S. Seamons, <a href="https://fq.math.ca/Scanned/4-2/elementary4-2.pdf">Problem B-89</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 4, No. 2 (1966), p. 190; <a href="https://www.fq.math.ca/Scanned/5-1/elementary5-1.pdf">A Close Approximation</a>, Solution to Problem B-89 by Douglas Lind, ibid., Vol. 5, No. 1 (1967), pp. 108-109.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F a(n) = Lucas(2*n)-1, with Lucas(n)=A000032(n).
%F a(n) = floor(r^(2*n)), where r = golden ratio = (1+sqrt(5))/2.
%F a(n) = floor(Fibonacci(5*n)/Fibonacci(3*n)). - _Gary Detlefs_, Mar 11 2011
%F a(n) = +4*a(n-1) -4*a(n-2) +1*a(n-3). - _Joerg Arndt_, Mar 11 2011
%F a(n) = A001519(2*n-1) + A001519(2*n+1) - 1. - _Reinhard Zumkeller_, Aug 09 2013
%F a(n) = 3*a(n) - a(n-1) + 1; a(n) = A004146(n) + 1, n>0. - _Richard R. Forberg_, Sep 04 2013
%F a(n) = 2*cosh(2*n*arcsinh(1/2)) - 1. - _Ilya Gutkovskiy_, Oct 31 2016
%F a(n) = floor(sqrt(5)*Fibonacci(2*n)), for n > 0 (Seamons, 1966). - _Amiram Eldar_, Feb 05 2022
%e G.f. = 1 + 2*x + 6*x^2 + 17*x^3 + 46*x^4 + 122*x^5 + 321*x^6 + 842*x^7 + ...
%p A005592:=-(2-2*z+z**2)/(z-1)/(z**2-3*z+1); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p # second Maple program:
%p F:= n-> (<<0|1>, <1|1>>^n)[1,2]:
%p a:= n-> F(2*n+1)+F(2*n-1)-1:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Nov 04 2016
%t Table[Fibonacci[2n+1]+Fibonacci[2n-1]-1,{n,30}] (* _Harvey P. Dale_, Aug 22 2011 *)
%t a[n_] := LucasL[2n]-1; Array[a, 30] (* _Jean-François Alcover_, Dec 09 2015 *)
%o (Sage) [lucas_number2(n,3,1)-1 for n in range(1,29)] # _Zerinvary Lajos_, Jul 06 2008
%o (Magma) [Fibonacci(2*n+1)+Fibonacci(2*n-1)-1: n in [1..30]]; // _Vincenzo Librandi_, Aug 23 2011
%o (PARI) a(n)=fibonacci(2*n+1)+fibonacci(2*n-1)-1 \\ _Charles R Greathouse IV_, Aug 23 2011
%o (Haskell)
%o a005592 n = a005592_list !! (n-1)
%o a005592_list = map (subtract 1) $
%o tail $ zipWith (+) a001519_list $ tail a001519_list
%o -- _Reinhard Zumkeller_, Aug 09 2013
%Y Equals A004146+1 and A005248+1.
%Y Bisection of A014217; the other bisection is A002878, which also bisects A000032.
%Y Cf. A000045, A065034.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Formulae and comments by _Clark Kimberling_, Nov 24 2010
%E a(0)=1 prepended by _Alois P. Heinz_, Nov 04 2016
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