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A267623 Binary representation of the middle column of the "Rule 187" elementary cellular automaton starting with a single ON (black) cell. 4
1, 10, 101, 1011, 10111, 101111, 1011111, 10111111, 101111111, 1011111111, 10111111111, 101111111111, 1011111111111, 10111111111111, 101111111111111, 1011111111111111, 10111111111111111, 101111111111111111, 1011111111111111111, 10111111111111111111 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Also, The binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283508.
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
LINKS
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
FORMULA
Conjectures from Colin Barker, Jan 19 2016 and Apr 16 2019: (Start)
a(n) = 11*a(n-1)-10*a(n-2) for n>2.
G.f.: (1-x+x^2) / ((1-x)*(1-10*x)).
(End)
Empirical: a(n) = (91*10^n - 10) / 90 for n>0. - Colin Barker, Mar 10 2017
It also appears that a(n) = floor(91*10^n/90). - Karl V. Keller, Jr., May 28 2022
MAPLE
# Rule 187: value in generation r and column c, where c=0 is the central one
r187 := proc(r::integer, c::integer)
option remember;
local up ;
if r = 0 then
if c = 0 then
1;
else
0;
end if;
else
# previous 3 bits
[procname(r-1, c+1), procname(r-1, c), procname(r-1, c-1)] ;
up := op(3, %)+2*op(2, %)+4*op(1, %) ;
# rule 187 = 10111011_2: {6, 2}->0, all others ->1
if up in {6, 2} then
0;
else
1 ;
end if;
end if;
end proc:
A267623 := proc(n)
b := [seq(r187(r, 0), r=0..n)] ;
add(op(-i, b)*2^(i-1), i=1..nops(b)) ;
A007088(%) ;
end proc:
smax := 30 ;
L := [seq(A267623(n), n=0..smax)] ; # R. J. Mathar, Apr 12 2019
MATHEMATICA
rule=187; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rows-k+1, rows+k-1}], {k, 1, rows}]; (* Truncated list of each row *) mc=Table[catri[[k]][[k]], {k, 1, rows}]; (* Keep only middle cell from each row *) Table[FromDigits[Take[mc, k]], {k, 1, rows}] (* Binary Representation of Middle Column *)
CROSSREFS
Sequence in context: A033128 A094945 A190480 * A283508 A261200 A175541
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jan 18 2016
EXTENSIONS
Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - N. J. A. Sloane, Jun 13 2022
STATUS
approved

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Last modified May 20 11:25 EDT 2024. Contains 372712 sequences. (Running on oeis4.)