|
|
A266841
|
|
Binary representation of the n-th iteration of the "Rule 69" elementary cellular automaton starting with a single ON (black) cell.
|
|
3
|
|
|
1, 10, 10100, 101011, 101010000, 1010101111, 1010101000000, 10101010111111, 10101010100000000, 101010101011111111, 101010101010000000000, 1010101010101111111111, 1010101010101000000000000, 10101010101010111111111111, 10101010101010100000000000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
LINKS
|
|
|
FORMULA
|
Conjectures from Colin Barker, Jan 05 2016 and Apr 18 2019: (Start)
a(n) = 10101*a(n-2)-1010100*a(n-4)+1000000*a(n-6) for n>5.
G.f.: (1+10*x-x^2+x^3-110000*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)*(1-100*x)*(1+100*x)).
(End)
Conjecture: a(n) = ( 10*100^n + 100^floor(n/2) - 11)/99 for odd n; a(n) = (100*100^n - 10^n)/99 for even n. - Karl V. Keller, Jr., Feb 28 2022
|
|
MATHEMATICA
|
rule=69; 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]]], {k, 1, rows}] (* Binary Representation of Rows *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|