A118101 Decimal representation of n-th iteration of the Rule 94 elementary cellular automaton starting with a single ON cell.
1, 7, 27, 119, 427, 1879, 6827, 30039, 109227, 480599, 1747627, 7689559, 27962027, 123032919, 447392427, 1968526679, 7158278827, 31496426839, 114532461227, 503942829399, 1832519379627, 8063085270359, 29320310074027, 129009364325719, 469124961184427
Offset: 0
Examples
From _Michael De Vlieger_, Oct 08 2015: (Start)
First 8 rows, representing ON cells as "1", OFF cells within the bounds of ON cells as "0", interpreted as a binary number at left, the decimal equivalent appearing at right:
1 = 1
111 = 7
1 1011 = 27
111 0111 = 119
1 1010 1011 = 427
111 0101 0111 = 1 879
1 1010 1010 1011 = 6 827
111 0101 0101 0111 = 30 039
1 1010 1010 1010 1011 = 109 227
(End)
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig. 10.
- Eric Weisstein's World of Mathematics, Rule 94
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- Index entries for sequences related to cellular automata
- Index entries for linear recurrences with constant coefficients, signature (0,17,0,-16).
Programs
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Mathematica
clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[94, {{1}, 0}, 24]] (* Michael De Vlieger, Oct 08 2015 *) -
Python
print([(11*4**n + 10)//6 - 2*0**abs(n-1) if n%2 else (5*4**n + 1)//3 - 0**n for n in range(50)]) # Karl V. Keller, Jr., Sep 10 2021
Formula
a(0)=1, a(1)=7, a(n odd) = (10+11*4^n)/6, a(n even) = (1+5*4^n)/3.
From Colin Barker, Oct 08 2015 and Apr 16 2019: (Start)
a(n) = (12-(-4)^n-8*(-1)^n+21*4^n)/12 for n>1.
a(n) = 17*a(n-2) - 16*a(n-4) for n>5.
G.f.: -(2*x+1)*(16*x^4-5*x-1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).
(End)