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A128588 A007318 * A128587. 12
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...

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.

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 + ...

MATHEMATICA

nn=20; 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 *)

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 */

CROSSREFS

Cf. A128587, A128586, A007318.

Cf. A006355, A055389.

Cf. A232642, A242593.

Cf. A038754.

Sequence in context: A227572 A080432 A094985 * A023613 A065795 A293077

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 November 17 19:40 EST 2017. Contains 294834 sequences.