A266976 Decimal representation of the n-th iteration of the "Rule 78" elementary cellular automaton starting with a single ON (black) cell.
1, 6, 28, 104, 464, 1696, 7488, 27264, 120064, 436736, 1922048, 6989824, 30756864, 111845376, 492126208, 1789558784, 7874084864, 28633071616, 125985619968, 458129670144, 2015770968064, 7330076819456, 32252339683328, 117281237499904, 516037451710464
Offset: 0
Links
- Robert Price, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
Programs
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Mathematica
rule=78; 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 08 2016 and Apr 18 2019: (Start)
a(n) = 2^(n-2)*((-2)^n+21*2^n-4)/3 = 2^(n-1)*A277954(n+1) for n>0.
a(n) = 2*a(n-1) + 16*a(n-2) - 32*a(n-3) for n>3.
G.f.: (1+4*x-16*x^3) / ((1-2*x)*(1-4*x)*(1+4*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