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1
1, 10, 18, 50, 66, 154, 170, 274, 290, 426, 498, 810, 866, 1226, 1302, 1770, 1846, 2290, 2446, 3154, 3306, 4234, 4398, 5394, 5550, 6474, 6766, 8082, 8338, 9858, 10206, 11882, 12186, 13954, 14354, 16282, 16690, 18730, 19254, 21882, 22398, 25082, 25770, 29042, 29786, 33658, 34342, 38034, 38790, 42514, 43354, 47434
COMMENTS
Also, total number of cubic ON cells after n generations in a three-dimensional structure in which A253090(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.
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.
+10
25
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
COMMENTS
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). 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
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; pp. 170-179.
LINKS
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.
FORMULA
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
EXAMPLE
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
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;
(End)
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)
MAPLE
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);
MATHEMATICA
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 *)
CROSSREFS
See A253090 for 9-celled neighborhood version.
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