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 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. 17
 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 (list; graph; refs; listen; history; text; internal format)
 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 REFERENCES Andreas M. Hinz, The Lichtenberg sequence, Fib. Quart., 55 (2017), 2-12. 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. A. M. Hinz, S. Klavžar, U. Milutinović, 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 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 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}]] (* Second program: *) LinearRecurrence[{2, 1, -2}, {1, 1, 4}, 40] (* Jean-François Alcover, Jan 08 2019 *) PROG (MAGMA) [(2^(n+1)-(1+(-1)^(n+1)))/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 CROSSREFS Cf. A000975, A135221. Row sums of A131086. Sequence in context: A286741 A298344 A285654 * A298415 A108122 A192800 Adjacent sequences:  A051046 A051047 A051048 * A051050 A051051 A051052 KEYWORD nonn,easy AUTHOR EXTENSIONS Edited and information added by Johannes W. Meijer, Jun 24 2011 STATUS approved

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Last modified May 14 11:06 EDT 2021. Contains 343882 sequences. (Running on oeis4.)