A081631 -id:A081631 - cp's OEIS frontend

A282124 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 430", based on the 5-celled von Neumann neighborhood.

1, 3, 3, 15, 11, 63, 43, 255, 171, 1023, 683, 4095, 2731, 16383, 10923, 65535, 43691, 262143, 174763, 1048575, 699051, 4194303, 2796203, 16777215, 11184811, 67108863, 44739243, 268435455, 178956971, 1073741823, 715827883, 4294967295, 2863311531, 17179869183

Offset: 0

Robert Price, Feb 06 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A282121, A282122, A282123.

Cf. A001045, A010684, A024036, A081631, A098646.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 430; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]

Conjectures from Colin Barker, Feb 07 2017: (Start)
a(n) = (-1 + 2*(-1)^n - (-1)^n*2^(1+n) + 2^(2+n)) / 3.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
G.f.: (1 + 3*x - 2*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
(End)
Conjectures from Paul Curtz, Jun 10 2019: (Start)
a(n) = A001045(n+1)*(period 2: repeat[1, 3]).
a(n+4) = a(n) + 10*A081631(n).
a(2*n+1) = 2^(2*n+2) -1.
a(n+2) = a(n) + A098646(n+1).
(End)

A081632 2*3^n-(-1)^n.

Original entry on oeis.org

1, 7, 17, 55, 161, 487, 1457, 4375, 13121, 39367, 118097, 354295, 1062881, 3188647, 9565937, 28697815, 86093441, 258280327, 774840977, 2324522935, 6973568801, 20920706407, 62762119217, 188286357655, 564859072961, 1694577218887

Offset: 0

Paul Barry, Mar 26 2003

Binomial transform of A081631. Inverse binomial transform of A081654.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3).

[2*3^n-(-1)^n: n in [0..40]]; // Vincenzo Librandi, Aug 10 2013

Total/@Partition[Riffle[2*3^Range[0,30],{-1,1}],2] (* or *) LinearRecurrence[{2,3},{1,7},30] (* Harvey P. Dale, May 24 2011 *)
CoefficientList[Series[(1 + 5 x) / ((1 - 3 x) (1 + x)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 10 2013 *)

G.f.: (1+5*x)/((1-3*x)(1+x)).
E.g.f. 2*exp(3*x)-exp(-x).
a(0)=1, a(1)=7, a(n)=2*a(n-1)+3*a(n-2) [ Harvey P. Dale, May 24 2011]

A081630 2-(-3)^n.

Original entry on oeis.org

1, 5, -7, 29, -79, 245, -727, 2189, -6559, 19685, -59047, 177149, -531439, 1594325, -4782967, 14348909, -43046719, 129140165, -387420487, 1162261469, -3486784399, 10460353205, -31381059607, 94143178829, -282429536479, 847288609445, -2541865828327, 7625597484989

Offset: 0

Paul Barry, Mar 26 2003

Binomial transform of A081629. Inverse binomial transform of A081631.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,3).

[2-(-3)^n: n in [0..30]]; // Vincenzo Librandi Aug 09 2013

LinearRecurrence[{-2, 3}, {1, 5}, 30] (* Harvey P. Dale, Mar 25 2013 *)
CoefficientList[Series[(1 + 7 x) / ((1 - x) (1 + 3 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 09 2013 *)

G.f.: (1+7*x)/((1-x)*(1+3*x)).

E.g.f. 2*exp(x)-exp(-3*x)

A176961 a(n) = (3*2^(n+1) - 8 - (-2)^n)/6.

Original entry on oeis.org

1, 2, 8, 12, 36, 52, 148, 212, 596, 852, 2388, 3412, 9556, 13652, 38228, 54612, 152916, 218452, 611668, 873812, 2446676, 3495252, 9786708, 13981012, 39146836, 55924052, 156587348, 223696212, 626349396, 894784852

Offset: 1

Roger L. Bagula, Apr 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

[(3*2^(n+1)-8-(-2)^n)/6:n in [1..40]]; // Vincenzo Librandi, Sep 15 2011

a[1] := 1;
a[n_] := a[n] = a[n - 1]/2 + Sqrt[(5 + 4*(-1)^(n - 1))]/2:
Table[2^(n - 1)*a[n], {n, 1, 30}]

a(n)=(3<<(n+1)-(-2)^n)\/6-1 \\ Charles R Greathouse IV, Sep 14 2011

a(n) - a(n-1) = A081631(n-2).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3).
G.f.: x*(1 + x + 2*x^2) / ((x-1)*(2*x+1)*(2*x-1)). - R. J. Mathar, Apr 30 2010
a(n) = 2^n - A084247(n-1). - Bruno Berselli, Sep 15 2011

cp's OEIS Frontend

A282124 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 430", based on the 5-celled von Neumann neighborhood.

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A081632 2*3^n-(-1)^n.

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A081630 2-(-3)^n.

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A176961 a(n) = (3*2^(n+1) - 8 - (-2)^n)/6.

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