A266068 Binary representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.
1, 100, 10, 1111001, 100, 11111110011, 1000, 111111111100111, 10000, 1111111111111001111, 100000, 11111111111111110011111, 1000000, 111111111111111111100111111, 10000000, 1111111111111111111111001111111, 100000000, 11111111111111111111111110011111111
Offset: 0
Examples
From _Michael De Vlieger_, Dec 21 2015: (Start)
First 8 rows, replacing leading zeros with ".", the row converted to its binary equivalent at right:
1 = 1
1 0 0 = 100
. . . 1 0 = 10
1 1 1 1 0 0 1 = 1111001
. . . . . . 1 0 0 = 100
1 1 1 1 1 1 1 0 0 1 1 = 11111110011
. . . . . . . . . 1 0 0 0 = 1000
1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 = 111111111100111
(End)
Links
- Robert Price, Table of n, a(n) for n = 0..999
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
- Index entries for linear recurrences with constant coefficients, signature (0,10011,0,-110010,0,100000).
Programs
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Mathematica
rule = 3; rows = 20; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}] -
Python
print([(10*100**n - 99*10**((n-1)//2) - 1)//9 if n%2 else 10**(n//2) for n in range(30)]) # Karl V. Keller, Jr., Aug 26 2021
Formula
G.f.: (1+100*x-10001*x^2+109901*x^3+10000*x^4-120000*x^5) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)*(1-10*x^2)). - Colin Barker, Dec 21 2015
a(n) = (10*100^n - 99*10^((n-1)/2) - 1)/9 for odd n; a(n) = 10^(n/2) for even n. - Karl V. Keller, Jr., Aug 26 2021
Comments