A071051 - OEIS

%I #58 Jan 09 2025 03:16:17

%S 1,3,4,7,4,8,8,15,4,8,8,16,8,16,16,31,4,8,8,16,8,16,16,32,8,16,16,32,

%T 16,32,32,63,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,8,16,16,32,16,

%U 32,32,64,16,32,32,64,32,64,64,127,4,8,8,16,8,16,16,32,8,16

%N Number of 1's in n-th row of triangle in A071035.

%C Number of ON cells at generation n of 1-D CA defined by Rule 126, starting with a single ON cell. - _N. J. A. Sloane_, Aug 09 2014

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

%H Robert Price, <a href="/A071051/b071051.txt">Table of n, a(n) for n = 0..1000</a>

%H A. J. Macfarlane, <a href="http://www.damtp.cam.ac.uk/user/ajm/Papers2016/GFsForCAsOfEvenRuleNo.ps">Generating functions for integer sequences defined by the evolution of cellular automata...</a>

%H 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 S. Wolfram, <a href="http://dx.doi.org/10.1103/RevModPhys.55.601">Statistical mechanics of cellular automata</a>, Rev. Mod. Phys., 55 (1983), 601--644.

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F a(2n) = a(n)+A036987(n); a(2n+1) = a(n)+2*2^A000120(n). - _Benoit Cloitre_, Sep 22 2003

%F a(n) = 2^(1+wt(n)) unless n is of the form 2^i-1 in which case we must subtract 1, where wt = A000120. - _N. J. A. Sloane_, Aug 09 2014

%F G.f.: 2*Product_{k>=0} (1+2*x^(2^k)) - Sum_{k>=0} x^(2^k-1). - _N. J. A. Sloane_, Aug 09 2014

%e [Contribution from _Omar E. Pol_, Dec 11 2010] (Start)

%e May be arranged in blocks of sizes 1, 1, 2, 4, 8, 16, 32, ...:

%e 1,

%e 3,

%e 4, 7,

%e 4, 8, 8, 15,

%e 4, 8, 8, 16, 8, 16, 16, 31,

%e 4, 8, 8, 16, 8, 16, 16, 32, 8, 16, 16, 32, 16, 32, 32, 63,

%e Last terms of rows give positive terms of A000225.

%e (End)

%t a[n_] := 2^(DigitCount[n, 2, 1]+1) - Boole[IntegerQ[Log[2, n+1]]];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Oct 02 2018, from 2nd formula *)

%Y Cf. A001316, A000120, A036987, A000225.

%K nonn,changed

%O 0,2

%A _Hans Havermann_, May 26 2002