A267044 Binary representation of the middle column of the "Rule 91" elementary cellular automaton starting with a single ON (black) cell.

1, 10, 100, 1000, 10001, 100010, 1000101, 10001010, 100010101, 1000101010, 10001010101, 100010101010, 1000101010101, 10001010101010, 100010101010101, 1000101010101010, 10001010101010101, 100010101010101010, 1000101010101010101, 10001010101010101010

Offset: 0

Author

Robert Price, Jan 09 2016

Keywords

• nonn

References

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

Links

• Robert Price, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• S. Wolfram, A New Kind of Science
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A267015, A267045.

Programs

Mathematica

rule=91; 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 *)

Formula

Conjectures from Colin Barker, Jan 10 2016: (Start)

a(n) = (-550+450*(-1)^n+9901*10^n)/9900 for n>1.

a(n) = 10*a(n-1)+a(n-2)-10*a(n-3) for n>4. [Typo corrected by Karl V. Keller, Jr., Mar 16 2022]

G.f.: (1-x^2+x^4) / ((1-x)*(1+x)*(1-10*x)).

(End)

Extensions

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - Michael De Vlieger, Jun 13 2022