A267941 Decimal representation of the n-th iteration of the "Rule 253" elementary cellular automaton starting with a single ON (black) cell.
1, 3, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311
Offset: 0
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. A060576.
Programs
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Mathematica
rule=253; 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[FromDigits[catri[[k]],2],{k,1,rows}] (* Decimal Representation of Rows *)
Formula
Conjectures from Colin Barker, Jan 23 2016 and Apr 16 2019: (Start)
a(n) = 5*a(n-1)-4*a(n-2) for n>3.
G.f.: (1-2*x+20*x^2-16*x^3) / ((1-x)*(1-4*x)).
(End)
Empirical a(n) = 2^(2*n+1) - 1 for n>1. - Colin Barker, Nov 26 2016
Comments