A253908 -id:A253908 - cp's OEIS frontend

A072272 Number of active cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 614", based on the 5-celled von Neumann neighborhood.

1, 5, 5, 17, 5, 25, 17, 61, 5, 25, 25, 85, 17, 85, 61, 217, 5, 25, 25, 85, 25, 125, 85, 305, 17, 85, 85, 289, 61, 305, 217, 773, 5, 25, 25, 85, 25, 125, 85, 305, 25, 125, 125, 425, 85, 425, 305, 1085, 17, 85, 85, 289, 85, 425, 289, 1037, 61, 305, 305, 1037, 217, 1085, 773, 2753

Offset: 0

Miklos Kristof, Jul 09 2002

Consider only the four nearest (N,S,E,W) neighbors of a cell together with the cell itself. In the next state, the state of a cell will change if an odd number of these five cells is ON. [Comment corrected by N. J. A. Sloane, Aug 25 2014]
Equivalently, a(n) is the number of ON cells at generation n of 2-D CA defined as follows: the neighborhood of a cell consists of the cell itself and the four adjacent E, W, N, S cells. A cell is ON iff an odd number of these cells was ON at the previous generation. - N. J. A. Sloane, Aug 20 2014. This is the odd-rule cellular automaton defined by OddRule 057 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
This is the Run Length Transform of A007483. - N. J. A. Sloane, Aug 25 2014
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g., 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product). - N. J. A. Sloane, Aug 25 2014
The partial sums are in A253908, in which the structure looks like an irregular stepped pyramid. - Omar E. Pol, Jan 29 2015
Rules 518, 550 and 582 also generate this sequence. - Robert Price, Mar 01 2016

			To illustrate a(0) = 1, a(1) = 5, a(2) = 5, a(3) = 17:
  ......................0
  .............0.......000
  .......0............0...0
  .0....000..0.0.0...00.0.00
  .......0............0...0
  .............0.......000
  ......................0
From _Omar E. Pol_, Jan 29 2015: (Start)
May be arranged into blocks of sizes A011782:
  1;
  5;
  5,17;
  5,25,17,61;
  5,25,25,85,17,85,61,217;
  5,25,25,85,25,125,85,305,17,85,85,289,61,305,217,773;
  5,25,25,85,25,125,85,305,25,125,125,425,85,425,305,1085,17,85,85,289,85,425,289,1037,
    61,305,305,1037,217,1085,773,2753;
So the right border gives A007483.
(End)
From _Omar E. Pol_, Mar 19 2015: (Start)
Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:
     1;
  .....
     5;
  .....
     5;
    17;
  ...........
     5,   25;
    17;
    61;
  ......................
     5,   25,  25,   85;
    17,   85;
    61;
   217;
  ...........................................
     5,   25,  25,   85,  25, 125,  85,  305;
    17,   85,  85,  289;
    61,  305;
   217;
   773;
  ..................................................................................
     5,   25,  25,   85,  25, 125,  85,  305, 25, 125, 125, 425, 85, 425, 305, 1085;
    17,   85,  85,  289,  85, 425, 289, 1037;
    61,  305, 305, 1037;
   217, 1085;
   773;
  2753;
  ...
Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).
It appears that the configuration of ON cells of T(s,r,k) is of the same kind as the configuration of ON cells of T(s+1,r,k).
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; pp. 170-179.

Alois P. Heinz, Table of n, a(n) for n = 0..8192
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
Nathan Epstein, Animation of CA generating A072272.
N. H. Packard and S. Wolfram, Two-Dimensional Cellular Automata, Journal of Statistical Physics, 38 (1985), 901-946.
N. J. A. Sloane, Illustration of first 15 generations.
N. J. A. Sloane, Illustration of first 28 generations.
N. J. A. Sloane, Illustration for a(15)=217.
N. J. A. Sloane, Illustration for a(31)=773.
N. J. A. Sloane, Illustration for a(63)=2753.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
S. Wolfram, A New Kind of Science.
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

Cf. A048883, A170878 (first differences), A253908 (partial sums).

See A253090 for 9-celled neighborhood version.

C:=f->subs({x=1,y=1},f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x,y).
OddCA:=proc(f,M) global C; local n,a,i,f2,g,p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1; g:=f2;
for n from 0 to M do a:=[op(a),C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i],i=1..nops(a))]);
end;
f:=1+1/x+x+1/y+y;
OddCA(f,100);
# N. J. A. Sloane, Aug 20 2014

Map[Function[Apply[Plus,Flatten[ #1]]], CellularAutomaton[{ 614, {2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},100]] (* N. J. A. Sloane, Apr 17 2010 *)
ArrayPlot /@ CellularAutomaton[{ 614, {2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},6] (* N. J. A. Sloane, Aug 25 2014 *)

a(0)=1; thereafter a(2t)=a(t), a(4t+1)=5*a(t), a(4t+3)=3*a(2t+1)+2*a(t). - N. J. A. Sloane, Jan 26 2015

Extended and edited by John W. Layman, Jul 17 2002
Minor edits by N. J. A. Sloane, Jan 07 2010
More terms from N. J. A. Sloane, Apr 17 2010

A245542 Partial sums of A160239.

Original entry on oeis.org

1, 9, 17, 41, 49, 113, 137, 249, 257, 321, 385, 577, 601, 793, 905, 1321, 1329, 1393, 1457, 1649, 1713, 2225, 2417, 3313, 3337, 3529, 3721, 4297, 4409, 5305, 5721, 7449, 7457, 7521, 7585, 7777, 7841, 8353, 8545, 9441, 9505, 10017, 10529, 12065

Offset: 0

N. J. A. Sloane, Jul 26 2014

Also, total number of cubic ON cells after n generations in a three-dimensional cellular automaton where A160239(n) gives the number of cubic ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. The structure looks like an irregular stepped pyramid. - Omar E. Pol, Jan 27 2015

N. J. A. Sloane, Table of n, a(n) for n = 0..16383
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Cf. A116520, A160239, A245540, A253908, A253767.

a245542 n = a245542_list !! n
a245542_list = scanl1 (+) a160239_list
-- Reinhard Zumkeller, Feb 13 2015

b[n_] := b[n] = Which[n == 1, 1, Mod[n, 2] == 0, b[n/2], Mod[n, 4] == 3, 2b[(n - 1)/2] + b[n - 2], True, 8b[(n - 1)/4]];
Join[{1}, 1 + 8 Accumulate[Array[b, 43]]] (* Jean-François Alcover, Oct 01 2018, after Omar E. Pol *)

a(n) = 1 + 8*A245540(n), n >= 1. - Omar E. Pol, Mar 07 2015

Offset changed to 0 by N. J. A. Sloane, Feb 06 2015

A269522 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 622", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 57, 69, 89, 101, 121, 133, 169, 205, 257, 309, 361, 333, 377, 381, 385, 461, 465, 509, 601, 653, 697, 781, 889, 845, 1097, 1077, 1353, 1205, 1305, 1349, 1345, 1357, 1321, 1437, 1553, 1589, 1601, 1709, 1729, 1813, 1937, 2069, 2361, 2181

Offset: 0

Robert Price, Feb 28 2016

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..300
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A007483, A072272, A170878, A253908.

rule=622; stages=300; ca=CellularAutomaton[ {rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}}, {{{1}},0}, stages]; (* Start with single black cell *) Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)

Also, total number of ON cells after n generations in a three-dimensional cellular automaton where A247666(n) gives the number of ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. An ON cell is an hexagonal prism of height 1. We start with a single ON cell. The structure looks like an irregular stepped pyramid, apparently with a like-hexagonal base.

Also, total number of cubic ON cells after n generations in a three-dimensional cellular automaton in which A253086(n) gives the number of cubic ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. The structure looks like an irregular stepped pyramid.

cp's OEIS Frontend

A072272 Number of active cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 614", based on the 5-celled von Neumann neighborhood.

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A245542 Partial sums of A160239.

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A269522 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 622", based on the 5-celled von Neumann neighborhood.

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A253767 Partial sums of A247666.

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A255150 Partial sums of A253086.

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