A118173 -id:A118173 - cp's OEIS frontend

A118174 Binary representation of n-th iteration of the Rule 188 elementary cellular automaton starting with a single black cell.

1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0

Offset: 0

Eric W. Weisstein, Apr 13 2006

			Triangle starts:
            1;
         0, 1, 1;
      0, 0, 1, 0, 1;
   0, 0, 0, 1, 1, 1, 1;
0, 0, 0, 0, 1, 1, 1, 0, 1;
...

Robert Price, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Rule 188
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A118173, decimal representation.
Cf. A265428, total number of ON cells in n-th iteration.
Cf. A265429, total number of ON cells after n iterations.
Cf. A265430, total number of OFF cells in n-th iteration.
Cf. A265431, total number of OFF cells after n iterations.

lim = 10; a = {}; Do[AppendTo[a, Table[0, k - 1]~Join~Take[#[[k]], k]], {k, Length@ #}] &@ CellularAutomaton[188, {{1}, 0}, lim]; Flatten@ a (* Michael De Vlieger, Dec 09 2015 *)

A265427 Binary representation of the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 11, 101, 1111, 11101, 110111, 1011101, 11110111, 111011101, 1101110111, 10111011101, 111101110111, 1110111011101, 11011101110111, 101110111011101, 1111011101110111, 11101110111011101, 110111011101110111, 1011101110111011101, 11110111011101110111

Offset: 0

Robert Price, Dec 08 2015

			From _Michael De Vlieger_, Dec 09 2015: (Start)
First 12 rows:
1
1 1
1 0 1
1 1 1 1
1 1 1 0 1
1 1 0 1 1 1
1 0 1 1 1 0 1
1 1 1 1 0 1 1 1
1 1 1 0 1 1 1 0 1
1 1 0 1 1 1 0 1 1 1
1 0 1 1 1 0 1 1 1 0 1
1 1 1 1 0 1 1 1 0 1 1 1
1 1 1 0 1 1 1 0 1 1 1 0 1
(End)

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

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Rule 188
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A118173, A118174.

rule = 188; rows = 20; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]

Conjectures from Colin Barker, Dec 09 2015 and Apr 18 2019: (Start)
a(n) = a(n-2) + 10000*a(n-4) - 10000*a(n-6) for n>5.
G.f.: (1+11*x+100*x^2+1100*x^3+1000*x^4-1000*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)*(1+100*x^2)).
(End)
Conjecture: a(n) = floor(11100*100^n/9999) + 10^n - floor(11100*10^n/9999)*10^n. - Karl V. Keller, Jr., Dec 15 2021

A334244 Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 950, based on the 4-celled von Neumann neighborhood starting with a single black cell.

Original entry on oeis.org

1, 15, 81, 959, 5185, 61391, 331857, 3929023, 21238849, 251457487, 1359286353, 16093279167, 86994326593, 1029969866703, 5567636901969, 65918071468991, 356328761726017, 4218756574015439, 22805040750465105, 270000420736988095, 1459522608029766721, 17280026927167238095

Offset: 1

Pietro Tiaraju Giavarina dos Santos, Apr 20 2020

nonn

a(n) is the decimal representation of the n-th step based on a simple initial condition, when a(1) = 1.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,64,0,1,0,-64).

Cf. A118171, A118173 (similar examples from elementary cellular automata).

Table[((1607 +311*(-1)^n)*8^n -1040 +130*(-1)^n -1008*Sqrt[2]*Cos[(2*n-1)*Pi/4] )/8190, {n, 25}] (* G. C. Greubel, May 29 2020 *)

Vec(x*(1 + 15*x + 17*x^2 - x^3) / ((1 - x)*(1 + x)*(1 - 8*x)*(1 + 8*x)*(1 + x^2)) + O(x^20)) \\ Colin Barker, Jun 10 2020

a(n) = (-1040 + 130*(-1)^n - (504 + 504*i)*(-i)^n - (504 - 504*i)*i^n + 1607*2^(3*n) + 311*(-1)^n*2^(3*n))/8190 where i = sqrt(-1).
G.f.: (1 + 15*x + 17*x^2 - x^3)/(1 - 64*x^2 - x^4 + 64*x^6).
From G. C. Greubel, May 29 2020: (Start)
a(n) = ( (1607 + 311*(-1)^n)*8^n - (1040 - 130*(-1)^n) - 1008*sqrt(2)*cos((2*n-1)*Pi/4) )/8190.
E.g.f.: (959*cosh(8*x) + 648*sinh(8*x) - 455*cosh(x) - 585*sinh(x) - 504*(cos(x) + sin(x)) )/4095.
(End)
a(n) = 64*a(n-2) + a(n-4) - 64*a(n-6) for n>6. - Colin Barker, Jun 10 2020

A334340 Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 434, based on the 4-celled von Neumann neighborhood starting with a single black cell.

Original entry on oeis.org

1, 11, 81, 699, 5441, 43723, 349201, 2796731, 22364481, 178965195, 1431573521, 11453377723, 91624653121, 733009857227, 5864040961041, 46912529804475, 375299632087361, 3002400290556619, 24019192622879761, 192153592724761787, 1537228586572923201, 12297829520450964171, 98382633680004977681

Offset: 1

Pietro Tiaraju Giavarina dos Santos, Apr 23 2020

Index entries for linear recurrences with constant coefficients, signature (8,16,-128,1,-8,-16,128).

Cf. A118171, A118173 (similar examples from elementary cellular automata).

a(n) = (1428 + 7*(-4)^n + 2278*(-1)^n + (1800 + 360*i)*(-i)^n + (1800 - 360*i)*i^n - 3*4^n + 85*8^n)/510 where i = sqrt(-1).

G.f.: (-1 - 3*x + 23*x^2 - 3*x^3 + 40*x^4 + 624*x^5 - 1856*x^6)/(-1 + 8*x + 16*x^2 - 128*x^3 + x^4 - 8*x^5 - 16*x^6 + 128*x^7).

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A118174 Binary representation of n-th iteration of the Rule 188 elementary cellular automaton starting with a single black cell.

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A265427 Binary representation of the n-th iteration of the "Rule 188" elementary cellular automaton starting with a single ON (black) cell.

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A334244 Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 950, based on the 4-celled von Neumann neighborhood starting with a single black cell.

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A334340 Decimal representation of n-th iteration of the one-dimensional cellular automaton defined by Rule 434, based on the 4-celled von Neumann neighborhood starting with a single black cell.

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