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, J. Int. Seq. 19 (2016), Art. 16.3.5.
Andreas M. Hinz, The Lichtenberg sequence, Fib. Quart., 55 (2017), 2-12.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 56. Book's website.
Andreas M. Hinz and El-Mehdi Mehiri, The Weighted Tower of Hanoi: Algebraic Structure, Phase Transitions, and Integer Sequences, arXiv:2605.16956 [math.CO], 2026. See p. 17.
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.
Stephen Wolfram, A New Kind of Science, 2002.
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) = 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) = 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}]
(* Alternative: *)
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
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
Edited and information added by Johannes W. Meijer, Jun 24 2011
STATUS
approved