A266247 Binary representation of the middle column of the "Rule 9" elementary cellular automaton starting with a single ON (black) cell.

1, 10, 101, 1010, 10101, 101011, 1010110, 10101101, 101011010, 1010110101, 10101101010, 101011010101, 1010110101010, 10101101010101, 101011010101010, 1010110101010101, 10101101010101010, 101011010101010101, 1010110101010101010, 10101101010101010101

Offset: 0

Author

Robert Price, Dec 25 2015

Keywords

• nonn
• easy

References

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

Links

• Robert Price, Table of n, a(n) for n = 0..999
• 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 (10,1,-10).

Crossrefs

Cf. A266243, A266248.

Programs

Mathematica

rule=9; 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 *)

Python

print([(100000*10**n//9 + 100001*10**n)//110000 for n in range(50)]) # Karl V. Keller, Jr., Dec 15 2021

Formula

From Colin Barker, Dec 28 2015 and Apr 14 2019: (Start)

a(n) = (-45000*(-1)^n + 1000009*10^n - 55000)/990000 for n > 3.

a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n > 6.

G.f.: (1 + x^5 - x^6) / ((1-x)*(1+x)*(1-10*x)).

(End)

a(n) = floor((100000*10^n/9 + 100001*10^n)/110000). - Karl V. Keller, Jr., Dec 15 2021