cp's OEIS Frontend

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.

%I A233411 #59 Feb 16 2025 08:33:21
%S A233411 0,0,2,4,12,24,56,112,240,480,992,1984,4032,8064,16256,32512,65280,
%T A233411 130560,261632,523264,1047552,2095104,4192256,8384512,16773120,
%U A233411 33546240,67100672,134201344,268419072,536838144,1073709056,2147418112,4294901760,8589803520
%N 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.
%C A233411 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
%C A233411 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.
%C A233411 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
%C A233411 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
%D A233411 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A233411 N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015
%H A233411 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A233411 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A233411 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A233411 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistinguishingNumber.html">Distinguishing Number</a>
%H A233411 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A233411 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A233411 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A233411 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-4).
%F A233411 G.f.: 2*x^2/( (1 - 2*x^2)*(1-2x) ).
%F A233411 a(n) = 2^n - 2^ceiling(n/2).
%F A233411 a(n) = 2*A032085(n) = 2*A122746(n-2) for n>=2. - _Alois P. Heinz_, Dec 09 2013
%e A233411 a(3) = 4 because we have: 000, 001, 110, 111.
%t A233411 nn=30;CoefficientList[Series[2x^2/(1-2x^2)/(1-2x),{x,0,nn}],x]
%t A233411 LinearRecurrence[{2,2,-4},{0,0,2},40] (* _Harvey P. Dale_, Sep 06 2015 *)
%o A233411 (PARI) a(n)=2^n-2^ceil(n/2) \\ _Charles R Greathouse IV_, Dec 09 2013
%Y A233411 Cf. A233533.
%K A233411 nonn,easy
%O A233411 0,3
%A A233411 _Geoffrey Critzer_, Dec 09 2013
%E A233411 Misplaced comment added by _Andrew Howroyd_, Sep 30 2017