|
|
A267681
|
|
Decimal representation of the n-th iteration of the "Rule 201" elementary cellular automaton starting with a single ON (black) cell.
|
|
3
|
|
|
1, 0, 21, 99, 471, 1935, 8031, 32319, 130431, 522495, 2094591, 8381439, 33544191, 134189055, 536829951, 2147368959, 8589770751, 34359279615, 137438298111, 549753978879, 2199020634111, 8796085682175, 35184361603071, 140737458995199, 562949911478271
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
The fact that only n cells to the left and right of the initially active cell are updated at step n (cf. A267679 for the binary representation) is contradictory to the usual treatment of a "Rule m" automaton, where all cells are updated depending on their neighborhood. See also the illustration of "Rule 201" on the MathWorld page. - M. F. Hasler, Jul 28 2018
|
|
REFERENCES
|
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 5*a(n-1)-20*a(n-3)+16*a(n-4) for n>4.
G.f.: (1-5*x+21*x^2+14*x^3-40*x^4) / ((1-x)*(1-2*x)*(1+2*x)*(1-4*x)).
(End)
Conjecture: a(n) = 2*4^n - (n%2*2 + [n]*5)*2^(n-1) - 1, where [n] = 1 iff n > 0; n%2 = 1 iff n is odd. - M. F. Hasler, Jul 28 2018
|
|
MATHEMATICA
|
rule=201; 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 *) Table[FromDigits[catri[[k]], 2], {k, 1, rows}] (* Decimal Representation of Rows *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|