A267872 Number of ON (black) cells in the n-th iteration of the "Rule 237" elementary cellular automaton starting with a single ON (black) cell.

1, 1, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset: 0

Author

Robert Price, Jan 21 2016

Keywords

• nonn
• easy

Comments

a(n) = A247328(n+1) for 2 <= n < 472, but a(472) = 945 differs from A247328(473) = 947. Furthermore, a(n) = A163985(n+1) for 2 <= n <= 1000. - Georg Fischer, Oct 22 2018

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. A163985, A267870.

Programs

Mathematica

rule=237; 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[Total[catri[[k]]],{k,1,rows}] (* Number of Black cells in stage n *)

Formula

Conjectures from Colin Barker, Jan 22 2016 and Apr 20 2019: (Start)

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

G.f.: (1-x+4*x^2-2*x^3) / (1-x)^2.

(End)