A153819 -id:A153819 - cp's OEIS frontend

A128588 Expansion of g.f. x*(1+x+x^2)/(1-x-x^2).

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

Offset: 1

Gary W. Adamson, Mar 11 2007

Previous name was: A007318 * A128587.
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
Row lengths of triangle A232642. - Reinhard Zumkeller, May 14 2015
a(n) is the number of binary strings of length n-1 avoiding the substrings 000 and 111. - Allan C. Wechsler, Feb 13 2025

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 and 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, and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016. See 1st column of Table 2 p. 11.
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
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).

Cf. A000045, A001622, A128587, A128586, A007318.
Cf. A006355, A055389.
Cf. A014217, A069403, A153819.
Cf. A232642, A242593.
Cf. A038754, A068922.
Main diagonal of A303696.

Concatenation([1], List([2..40], n-> 2*Fibonacci(n))); # G. C. Greubel, Jul 10 2019

a128588 n = a128588_list !! (n-1)
a128588_list = 1 : cows where
                   cows = 2 : 4 : zipWith (+) cows (tail cows)
-- Reinhard Zumkeller, May 14 2015

[1] cat [2*Fibonacci(n): n in [2..40]]; // G. C. Greubel, Jul 10 2019

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

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 *)

{a(n) = if( n<2, n==1, 2 * fibonacci(n))}; /* Michael Somos, Jul 18 2015 */

[1]+[2*fibonacci(n) for n in (2..40)] # G. C. Greubel, Jul 10 2019

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
E.g.f.: 4*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5) - x. - Stefano Spezia, Feb 19 2023

New name from Joerg Arndt, Feb 16 2024

A069403 a(n) = 2Fibonacci(2n+1) - 1.

Original entry on oeis.org

1, 3, 9, 25, 67, 177, 465, 1219, 3193, 8361, 21891, 57313, 150049, 392835, 1028457, 2692537, 7049155, 18454929, 48315633, 126491971, 331160281, 866988873, 2269806339, 5942430145, 15557484097, 40730022147, 106632582345, 279167724889, 730870592323, 1913444052081

Offset: 0

R. H. Hardin, Mar 22 2002

Half the number of n X 3 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Indices of A017245 = 9*n + 7 = 7, 16, 25, 34, for submitted A153819 = 16, 34, 88,. A153819(n) = 9*a(n) + 7 = 18*F(2*n+1) -2; F(n) = Fibonacci = A000045, 2's = A007395. Other recurrence: a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3). - Paul Curtz, Jan 02 2009

Colin Barker, Table of n, a(n) for n = 0..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 41, 56.
J. Hietarinta and C.-M. Viallet, Singularity confinement and chaos in discrete systems, Physical Review Letters 81 (1998), pp. 326-328.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A000045, A084707.
Cf. 1 X n A000225, 2 X n A016269, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.
Equals A052995 - 1.
Bisection of A001595, A062114, A066983.

List([0..30], n-> 2*Fibonacci(2*n+1)-1); # G. C. Greubel, Jul 11 2019

[2*Fibonacci(2*n+1)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

a[n_]:= a[n] = 3a[n-1] - 3a[n-3] + a[n-4]; a[0] = 1; a[1] = 3; a[2] = 9; a[3] = 25; Table[ a[n], {n, 0, 30}]
Table[2*Fibonacci[2*n+1]-1, {n,0,30}] (* G. C. Greubel, Apr 22 2018 *)
LinearRecurrence[{4,-4,1},{1,3,9},30] (* Harvey P. Dale, Sep 22 2020 *)

a(n) = 2*fibonacci(2*n+1)-1 \\ Charles R Greathouse IV, Jun 11 2015

Vec((1-x+x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 02 2016

[2*fibonacci(2*n+1)-1 for n in (0..30)] # G. C. Greubel, Jul 11 2019

a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 25; a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
a(n) = 3*a(n-1) - a(n-2) + 1 for n>1, a(1) = 3, a(0) = 0. - Reinhard Zumkeller, May 02 2006
From R. J. Mathar, Feb 23 2009: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
G.f.: (1-x+x^2)/((1-x)*(1-3*x+x^2)). (End)
a(n) = 1 + 2*Sum_{k=0..n} Fibonacci(2*k) = 1+2*A027941(n). - Gary Detlefs, Dec 07 2010
a(n) = (2^(-n)*(-5*2^n -(3-sqrt(5))^n*(-5+sqrt(5)) +(3+sqrt(5))^n*(5+sqrt(5))))/5. - Colin Barker, Nov 02 2016

Simpler definition from Vladeta Jovovic, Mar 19 2003

A153981 a(n) = 36Fibonacci(2n+1) - 4.

Original entry on oeis.org

32, 68, 176, 464, 1220, 3200, 8384, 21956, 57488, 150512, 394052, 1031648, 2700896, 7071044, 18512240, 48465680, 126884804, 332188736, 869681408, 2276855492, 5960885072, 15605799728, 40856514116, 106963742624, 280034713760, 733140398660

Offset: 0

Paul Curtz, Jan 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (4, -4, 1).

[36*Fibonacci(2*n+1)-4: n in [0..30]]; // Vincenzo Librandi, Aug 07 2011

36*Fibonacci[2*Range[0,30]+1]-4 (* or *) LinearRecurrence[{4,-4,1},{32,68,176},30] (* Harvey P. Dale, Jan 26 2013 *)

a(n) = 4*A153873(n) = 2*A153819(n).
a(n) = 5 (mod 9) = A010716(n) (mod 9).
a(n) = 3*a(n-1) - a(n-2) + 4.
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
G.f.: 4*(8 - 15*x + 8x^2)/((1-x)*(1 -3*x +x^2)). - R. J. Mathar, Jan 23 2009

Edited and extended by R. J. Mathar and N. J. A. Sloane, Jan 23 2009

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A128588 Expansion of g.f. x*(1+x+x^2)/(1-x-x^2).

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A069403 a(n) = 2*Fibonacci(2*n+1) - 1.

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A153981 a(n) = 36*Fibonacci(2*n+1) - 4.

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A069403 a(n) = 2Fibonacci(2n+1) - 1.

A153981 a(n) = 36Fibonacci(2n+1) - 4.