A266590 - OEIS

%I #30 Jul 06 2023 12:36:13

%S 1,2,14,65,56,1799,224,31775,896,520319,3584,8372735,14336,134154239,

%T 57344,2147229695,229376,34358722559,917504,549751750655,3670016,

%U 8796076769279,14680064,140737423343615,58720256,2251799553638399,234881024,36028795978776575

%N Decimal representation of the n-th iteration of the "Rule 37" elementary cellular automaton starting with a single ON (black) cell.

%H Robert Price, <a href="/A266590/b266590.txt">Table of n, a(n) for n = 0..1000</a>

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

%H Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 55.

%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>

%F Conjectures from _Colin Barker_, Jan 02 2016 and Apr 18 2019: (Start)

%F a(n) = 21*a(n-2)-84*a(n-4)+64*a(n-6) for n>7.

%F G.f.: (1+2*x-7*x^2+23*x^3-154*x^4+602*x^5+160*x^6-672*x^7) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).

%F (End)

%F Conjecture: a(n) = 2*4^n - 31*2^(n-2) - 1 for odd n > 1; a(n) = 7*2^(n-1) for even n > 1. - _Karl V. Keller, Jr._, Oct 06 2021

%t rule=37; 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 *)

%Y Cf. A266588, A266589.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 01 2016