A118108 Decimal representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.
1, 7, 17, 119, 273, 1911, 4369, 30583, 69905, 489335, 1118481, 7829367, 17895697, 125269879, 286331153, 2004318071, 4581298449, 32069089143, 73300775185, 513105426295, 1172812402961, 8209686820727, 18764998447377, 131354989131639, 300239975158033
Offset: 0
Examples
From _Michael De Vlieger_, Oct 07 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 0001 = 17
111 0111 = 119
1 0001 0001 = 273
111 0111 0111 = 1911
1 0001 0001 0001 = 4369
111 0111 0111 0111 = 30583
1 0001 0001 0001 0001 = 69905
(End)
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..1660
- A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig 8.
- Eric Weisstein's World of Mathematics, Rule 54
- 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
-
Mathematica
clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[54, {{1}, 0}, 27]] (* or *) Table[If[EvenQ@ n, (4^(n + 2) - 1), 7 (4^(n + 1) - 1)]/15, {n, 0, 27}] (* Michael De Vlieger, Oct 07 2015 *) -
Python
print([(16+12*(n%2))*4**n//15 for n in range(30)]) # Karl V. Keller, Jr., Aug 04 2021
Formula
a(n) = 7*(4^(n+1)-1)/15 for n odd; a(n) = (4^(n+2)-1)/15 for n even.
From Colin Barker, Oct 08 2015 and Apr 16 2019: (Start)
a(n) = 17*a(n-2) - 16*a(n-4) for n>3.
G.f.: (7*x+1) / ((x-1)*(x+1)*(4*x-1)*(4*x+1)).
(End)
a(n) = floor((16+12*(n mod 2))*4^n/15). - Karl V. Keller, Jr., Aug 04 2021
Extensions
a(23)-a(24) from Michael De Vlieger, Oct 07 2015
Comments