OFFSET
0,3
COMMENTS
Also, the number of non-symmetric compositions of n+1, e.g. 4 can be written 1+3, 3+1, 1+1+2, or 2+1+1 (but not 4, 2+2, 1+2+1 or 1+1+1+1). - Henry Bottomley, Jun 27 2005
If we examine the set of all binary words with infinite length we find that the average length of the shortest prefix which satisfies the above conditions is 4.
a(n) is also the number of minimum distinguishing (2-)labelings of the path graph P_n for n > 1. - Eric W. Weisstein, Oct 16 2014
Also, the decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Apr 22 2017
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
LINKS
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, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Eric Weisstein's World of Mathematics, Distinguishing Number
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).
FORMULA
G.f.: 2*x^2/( (1 - 2*x^2)*(1-2x) ).
a(n) = 2^n - 2^ceiling(n/2).
EXAMPLE
a(3) = 4 because we have: 000, 001, 110, 111.
MATHEMATICA
nn=30; CoefficientList[Series[2x^2/(1-2x^2)/(1-2x), {x, 0, nn}], x]
LinearRecurrence[{2, 2, -4}, {0, 0, 2}, 40] (* Harvey P. Dale, Sep 06 2015 *)
PROG
(PARI) a(n)=2^n-2^ceil(n/2) \\ Charles R Greathouse IV, Dec 09 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Geoffrey Critzer, Dec 09 2013
EXTENSIONS
Misplaced comment added by Andrew Howroyd, Sep 30 2017
STATUS
approved