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 A128588 A007318 * A128587. 17
 1, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n)/a(n-1) tends to phi, 1.618... = A001622. Regardless of initial two terms, any linearly recurring sequence with signature (1,1) will yield an a(n)/a(n+1) ratio tending to phi (the Golden Ratio). - Harvey P. Dale, Mar 29 2017 Apart from the initial term, double the Fibonacci numbers. O.g.f.: x*(1+x+x^2)/(1-x-x^2). a(n) gives the number of binary strings of length n-1 avoiding the substrings 000 and 111. a(n) also gives the number of binary strings of length n-1 avoiding the substrings 010 and 101. - Peter Bala, Jan 22 2008 From A014217=1,1,2,4,6,. Which leads to A153819=16,34,88. Inverse binomial transform of A069403=1,3,9,25,67. - Paul Curtz, Jan 03 2009 Variation on "Narayana's Cows". One cow at step n=1. At any subsequent step any cow generates another one but after two steps dies. The sequence gives the total number of cows at any steps. - Paolo P. Lava, Oct 07 2009 Row lengths of triangle A232642. - Reinhard Zumkeller, May 14 2015 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..2500 J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms. Andrei Asinowski, Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16. Elena Barcucci, Antonio Bernini, Stefano Bilotta, Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016. See 1st column of Table 2 p. 11. P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 52. B. Winterfjord, Binary strings and substring avoidance. Index entries for linear recurrences with constant coefficients, signature (1,1). FORMULA G.f.: x*(1+x+x^2)/(1-x-x^2). Binomial transform of A128587; a(n+2) = a(n+1) + a(n), n>3. a(n) = A068922(n-1), n>2. - R. J. Mathar, Jun 14 2008 For n > 1: a(n+1) = a(n) + if a(n) odd then max{a(n),a(n-1)} else min{a(n),a(n-1)}, see also A038754. - Reinhard Zumkeller, Oct 19 2015 EXAMPLE a(4) = 6 = 1*1 + 3*1 + 3*1 + 1*(-1); where A128587 = (1, 1, 1, -1, 3, -5, 9, ...). G.f. = x + 2*x^2 + 4*x^3 + 6*x^4 + 10*x^5 + 16*x^6 + 26*x^7 + 42*x^8 + ... MAPLE a:= n-> `if`(n<2, n, 2*(<<0|1>, <1|1>>^n)[1, 2]): seq(a(n), n=1..50);  # Alois P. Heinz, Apr 28 2018 MATHEMATICA nn=40; a=(1-x^3)/(1-x); b=x*(1-x^2)/(1-x); CoefficientList[Series[a^2 /(1-b^2), {x, 0, nn}], x]  (* Geoffrey Critzer, Sep 01 2012 *) LinearRecurrence[{1, 1}, {1, 2, 4}, 40] (* Harvey P. Dale, Mar 29 2017 *) Join[{1}, 2*Fibonacci[Range[2, 40]]] (* G. C. Greubel, Jul 10 2019 *) PROG (Haskell) a128588 n = a128588_list !! (n-1) a128588_list = 1 : cows where                    cows = 2 : 4 : zipWith (+) cows (tail cows) -- Reinhard Zumkeller, May 14 2015 (PARI) {a(n) = if( n<2, n==1, 2 * fibonacci(n))}; /* Michael Somos, Jul 18 2015 */ (Magma)  cat [2*Fibonacci(n): n in [2..40]]; // G. C. Greubel, Jul 10 2019 (Sage) +[2*fibonacci(n) for n in (2..40)] # G. C. Greubel, Jul 10 2019 (GAP) Concatenation(, List([2..40], n-> 2*Fibonacci(n))); # G. C. Greubel, Jul 10 2019 CROSSREFS Cf. A000045, A001622, A128587, A128586, A007318. Cf. A006355, A055389. Cf. A232642, A242593. Cf. A038754. Main diagonal of A303696. Sequence in context: A080432 A094985 A336662 * A023613 A306293 A307795 Adjacent sequences:  A128585 A128586 A128587 * A128589 A128590 A128591 KEYWORD nonn,easy AUTHOR Gary W. Adamson, Mar 11 2007 STATUS approved

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Last modified October 2 08:34 EDT 2022. Contains 357191 sequences. (Running on oeis4.)