A265688 Binary representation of the n-th iteration of the "Rule 190" elementary cellular automaton starting with a single ON (black) cell.

1, 111, 11101, 1110111, 111011101, 11101110111, 1110111011101, 111011101110111, 11101110111011101, 1110111011101110111, 111011101110111011101, 11101110111011101110111, 1110111011101110111011101, 111011101110111011101110111, 11101110111011101110111011101

Offset: 0

Author

Robert Price, Dec 13 2015

Keywords

• nonn
• easy

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

• Robert Price, Table of n, a(n) for n = 0..499
• Eric Weisstein's World of Mathematics, Rule 190
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata
• Index entries for linear recurrences with constant coefficients, signature (100,1,-100).

Crossrefs

Cf. A037576 (decimal), A118111.

Programs

Mathematica

rule = 190; 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([11100*100**n//9999 for n in range(30)]) # Karl V. Keller, Jr., Aug 10 2021

Formula

From Colin Barker, Dec 13 2015 and Apr 18 2019: (Start)

a(n) = (-165*(-1)^n+37*100^(n+1)-202)/3333.

a(n) = (37*100^(n+1)-367)/3333 for n even.

a(n) = (37*100^(n+1)-37)/3333 for n odd.

a(n) = 100*a(n-1) + a(n-2) - 100*a(n-3) for n>2.

G.f.: (1+11*x) / ((1-x)*(1+x)*(1-100*x)).

(End)

a(n) = floor(11100*100^n/9999). - Karl V. Keller, Jr., Aug 10 2021