A233411 The number of length n binary words with some prefix which contains two more 1's than 0's or two more 0's than 1's.

0, 0, 2, 4, 12, 24, 56, 112, 240, 480, 992, 1984, 4032, 8064, 16256, 32512, 65280, 130560, 261632, 523264, 1047552, 2095104, 4192256, 8384512, 16773120, 33546240, 67100672, 134201344, 268419072, 536838144, 1073709056, 2147418112, 4294901760, 8589803520

Offset: 0

Author

Geoffrey Critzer, Dec 09 2013

Keywords

• nonn
• easy

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

Examples

			a(3) = 4 because we have: 000, 001, 110, 111.
		

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 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 (2,2,-4).

Crossrefs

Cf. A233533.

Programs

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 *)

PARI

a(n)=2^n-2^ceil(n/2) \\ Charles R Greathouse IV, Dec 09 2013

Formula

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

a(n) = 2^n - 2^ceiling(n/2).

a(n) = 2*A032085(n) = 2*A122746(n-2) for n>=2. - Alois P. Heinz, Dec 09 2013

Extensions

Misplaced comment added by Andrew Howroyd, Sep 30 2017