

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.


14



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
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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 xaxis, from the origin to the right edge, of the nth stage of growth of the twodimensional 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 5celled von Neumann neighborhood.  Robert Price, Jan 02 2017


REFERENCES

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), 212.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [From Wolfdieter Lang, Oct 18 2010]
A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi  Myths and Maths, Birkhäuser 2013. See page 56. Book's website
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 5Neighbor 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
G.f.: (1x+x^2)/((1x^2)*(12*x)); E.g.f.: exp(2*x)  sinh(x).  Paul Barry, Sep 19 2003
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, 2k).  Paul Barry, May 27 2003
a(n) = (Sum_{k=0..n} binomial(n,k) + (1)^(nk))1.  Paul Barry, Jul 21 2003
a(n) = sum{k=0..n, sum{j=0..nk, if(mod(jk, 2)=0, binomial(nk, j), 0}}.  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(n1) + a(n2)  2*a(n3), a(0)=1=a(1), a(2)=4. Observed by G. Detlefs. See the W. Lang link. [Wolfdieter Lang, Oct 18 2010]
a(n) = 3*a(n1)2*a(n2) +3*(1)^n. [Gary Detlefs, Dec 21 2010]
a(n) = 3* A000975(n1) + 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)=1k!*(x^(k+1))/((2k+1)!*(2^k) 2*(((2k+1)!*(2^k))^2)/( (2k+1)!*(2^(k+1))(x^k)*((k+1)!)/Q(k+1))); (continued fraction).  Sergei N. Gladkovskii, Nov 16 2011


MAPLE

A051049 := proc(n): (2^(n+1)(1+(1)^(n+1)))/2 end: seq(A051049(n), n=0..30); # Johannes W. Meijer, Jun 24 2011


MATHEMATICA

b[n_?EvenQ] := 2^(n  1)  1; b[n_?OddQ] := 2^(n  1); Table[b[n], {n, 50}]]


PROG

(MAGMA) [(2^(n+1)(1+(1)^(n+1)))/2: n in [0..40]]; // Vincenzo Librandi, Aug 14 2011
(PARI) a(n)=2^(n1)(n%2==0) \\ Charles R Greathouse IV, Mar 22 2013


CROSSREFS

Cf. A000975, A135221. Row sums of A131086.
Sequence in context: A240736 A286741 A285654 * A108122 A192800 A027609
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



