A051049 Number of moves needed to solve an (n+1)-ring baguenaudier if two simultaneous moves of the two end rings are counted as one.

1, 1, 4, 7, 16, 31, 64, 127, 256, 511, 1024, 2047, 4096, 8191, 16384, 32767, 65536, 131071, 262144, 524287, 1048576, 2097151, 4194304, 8388607, 16777216, 33554431, 67108864, 134217727, 268435456, 536870911, 1073741824

Offset

0,3

Comments

Might be called the "Purkiss sequence", after Henry John Purkiss who in 1865 found that this is the number of moves for the accelerated Chinese Rings puzzle (baguenaudier). [Email from Andreas M. Hinz, Feb 15 2017, who also pointed out that there was an error in the definition in this entry]. - N. J. A. Sloane, Feb 18 2017

The row sums of triangle A166692. - Paul Curtz, Oct 20 2009

The inverse binomial transform equals (-1)^n*A062510(n) with an extra leading term 1. - Paul Curtz, Oct 20 2009

This is the sequence A(1,1;1,2;1) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

Also, the decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by Rules 261, 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 37, and 381, based on the 5-celled von Neumann neighborhood. - Robert Price, Jan 02 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Andreas M. Hinz, The Lichtenberg sequence, Fib. Quart., 55 (2017), 2-12.

A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 56. Book's website

Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [From Wolfdieter Lang, Oct 18 2010]

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.

Eric Weisstein's World of Mathematics, Baguenaudier

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Wolfram Research, Wolfram Atlas of Simple Programs

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Formula

a(n) = (2^(n+1) - (1 + (-1)^(n+1)))/2. - Paul Barry, Apr 24 2003

a(n+2) = a(n+1) + 2*a(n) + 1, a(0)=a(1)=1. - Paul Barry, May 01 2003

From Paul Barry, Sep 19 2003: (Start)

G.f.: (1 - x + x^2)/((1 - x^2)*(1 - 2*x));

e.g.f.: exp(2*x) - sinh(x). (End)

a(n) = ((Sum_{k=0..n} 2^k) + (-1)^n)/2 = (A000225(n+1) + (-1)^n)/2. - Paul Barry, May 27 2003

(a(n+1) - a(n))/3 = A001045(n). - Paul Barry, May 27 2003

a(n) = Sum_{k=0..floor(n/2)} binomial(n+1, 2*k). - Paul Barry, May 27 2003

a(n) = (Sum_{k=0..n} binomial(n,k) + (-1)^(n-k)) - 1. - Paul Barry, Jul 21 2003

a(n) = Sum_{k=0..n} Sum_{j=0..n-k, (j-k) mod 2 = 0} binomial(n-k, j). - Paul Barry, Jan 25 2005

Row sums of triangle A135221. - Gary W. Adamson, Nov 23 2007

a(n) = A001045(n+1) + A000975(n+1) - A000079(n). - Paul Curtz, Oct 20 2009

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0) = a(1) = 1, a(2) = 4. Observed by G. Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010

a(n) = 3*a(n-1) - 2*a(n-2) + 3*(-1)^n. - Gary Detlefs, Dec 21 2010

a(n) = 3* A000975(n-1) + 1, n > 0. - Gary Detlefs, Dec 21 2010

a(n+2) = A001969(2^n+1) + A000069(2^n); evil + odious. - Johannes W. Meijer, Jun 24 2011, Jun 26 2011

E.g.f.: exp(2x) - sinh(x) = Q(0); Q(k) = 1 - k!*x^(k+1)/((2*k + 1)!*2^k - 2*(((2*k + 1)!*2^k)^2)/((2*k + 1)!*2^(k+1) - x^k*(k + 1)!/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 16 2011

a(n) = Sum_{k=0..n} Sum_{i=0..n} C(k-1,i). - Wesley Ivan Hurt, Sep 21 2017

a(n) = A000975(n+1) - A001045(n). - Yuchun Ji, Jul 08 2018

a(n) = A026147(2^(n-1)) for n > 0. - Chunqing Liu, Dec 18 2022

Maple

A051049:= proc(n): 2^n -(1-(-1)^n)/2 end: seq(A051049(n), n=0..40); # Johannes W. Meijer, Jun 24 2011

Mathematica

a[n_?EvenQ]:= 2^(n-1) -1; a[n_?OddQ]:= 2^(n-1); Table[a[n], {n, 50}]
LinearRecurrence[{2, 1, -2}, {1, 1, 4}, 40] (* Jean-François Alcover, Jan 08 2019 *)

Prog

(Magma) [2^n -(1-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Aug 14 2011
(PARI) a(n)=2^(n-1)-(n%2==0) \\ Charles R Greathouse IV, Mar 22 2013
(SageMath) [2^n -(n%2) for n in range(41)] # G. C. Greubel, Apr 23 2023

Crossrefs

Cf. A000069, A000079, A000225, A000975, A001045.

Cf. A001969, A026147, A062510, A135221.

Row sums of A131086.

Row sums of A166692.

Sequence in context: A286741 A298344 A285654 * A298415 A108122 A192800

Adjacent sequences: A051046 A051047 A051048 * A051050 A051051 A051052

Keyword

nonn,easy

Author

Eric W. Weisstein

Extensions

Edited and information added by Johannes W. Meijer, Jun 24 2011

Status

approved