cp's OEIS Frontend

A265721 Decimal representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

%I A265721 #31 Feb 16 2025 08:33:28
%S A265721 1,0,4,99,16,1935,64,32319,256,522495,1024,8381439,4096,134189055,
%T A265721 16384,2147368959,65536,34359279615,262144,549753978879,1048576,
%U A265721 8796085682175,4194304,140737458995199,16777216,2251799696244735,67108864,36028796549201919,268435456
%N A265721 Decimal representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
%C A265721 Rule 33 also generates this sequence.
%D A265721 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A265721 Robert Price, <a href="/A265721/b265721.txt">Table of n, a(n) for n = 0..999</a>
%H A265721 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265721 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265721 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A265721 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,21,0,-84,0,64).
%F A265721 From _Colin Barker_, Dec 14 2015 and Apr 16 2019: (Start)
%F A265721 a(n) = 21*a(n-2) - 84*a(n-4) + 64*a(n-6) for n>5.
%F A265721 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 A265721 (End)
%F A265721 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 A265721 From _Michael De Vlieger_, Dec 14 2015: (Start)
%e A265721 First 8 rows, replacing leading zeros with ".", the row converted to its binary (A265720), then decimal equivalent at right:
%e A265721               1                ->               1  =     1
%e A265721             . . 0              ->               0  =     0
%e A265721           . . 1 0 0            ->             100  =     4
%e A265721         1 1 0 0 0 1 1          ->         1100011  =    99
%e A265721       . . . . 1 0 0 0 0        ->           10000  =    16
%e A265721     1 1 1 1 0 0 0 1 1 1 1      ->     11110001111  =  1935
%e A265721   . . . . . . 1 0 0 0 0 0 0    ->         1000000  =    64
%e A265721 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1  -> 111111000111111  = 32319
%e A265721 (End)
%t A265721 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 A265721 (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 A265721 Cf. A265718, A265720, A059841, A056830, A000975, A265722, A128918, A265723, A265724.
%K A265721 nonn,easy
%O A265721 0,3
%A A265721 _Robert Price_, Dec 14 2015