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Decimal representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
5

%I #31 Feb 16 2025 08:33:28

%S 1,0,4,99,16,1935,64,32319,256,522495,1024,8381439,4096,134189055,

%T 16384,2147368959,65536,34359279615,262144,549753978879,1048576,

%U 8796085682175,4194304,140737458995199,16777216,2251799696244735,67108864,36028796549201919,268435456

%N Decimal representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

%C Rule 33 also generates this sequence.

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H Robert Price, <a href="/A265721/b265721.txt">Table of n, a(n) for n = 0..999</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,21,0,-84,0,64).

%F From _Colin Barker_, Dec 14 2015 and Apr 16 2019: (Start)

%F a(n) = 21*a(n-2) - 84*a(n-4) + 64*a(n-6) for n>5.

%F G.f.: (1-17*x^2+99*x^3+16*x^4-144*x^5) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).

%F (End)

%F a(n) = 2*4^n - 7*2^(n-1) - 1 for odd n; a(n) = 2^n for even n. - _Karl V. Keller, Jr._, Aug 24 2021

%e From _Michael De Vlieger_, Dec 14 2015: (Start)

%e First 8 rows, replacing leading zeros with ".", the row converted to its binary (A265720), then decimal equivalent at right:

%e 1 -> 1 = 1

%e . . 0 -> 0 = 0

%e . . 1 0 0 -> 100 = 4

%e 1 1 0 0 0 1 1 -> 1100011 = 99

%e . . . . 1 0 0 0 0 -> 10000 = 16

%e 1 1 1 1 0 0 0 1 1 1 1 -> 11110001111 = 1935

%e . . . . . . 1 0 0 0 0 0 0 -> 1000000 = 64

%e 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 -> 111111000111111 = 32319

%e (End)

%t rule = 1; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]],2], {k,1,rows}]

%o (Python) print([2*4**n - 7*2**(n-1) - 1 if n%2 else 2**n for n in range(50)]) # _Karl V. Keller, Jr._, Aug 24 2021

%Y Cf. A265718, A265720, A059841, A056830, A000975, A265722, A128918, A265723, A265724.

%K nonn,easy,changed

%O 0,3

%A _Robert Price_, Dec 14 2015