proposed

approved

editing

proposed

Cf. A286967, A286968, A286969.

Robert Price, <a href="/A286943/a286943.tmp.txt">Diagrams of first 20 stages</a>

Robert Price, <a href="/A286943/a286943.tmp.txt">Diagrams of first 20 stages</a>

Robert Price, <a href="/A286943/b286943.txt">Table of n, a(n) for n = 0..126</a>

Robert Price, <a href="/A286943/a286943.tmp.txt">Diagrams of first 20 stages</a>

allocated for Robert PriceBinary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 233", based on the 5-celled von Neumann neighborhood.

1, 1, 110, 111, 10100, 111, 11000, 10101111, 10000, 11111, 10001100000, 11111111, 1000001000000, 1101111111, 100000110000000, 110011111111, 10000011100000000, 11110111111111, 1000000011000000000, 1111001111111111, 100000010110000000000, 1101011111111111

0,3

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

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

N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015

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

S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>

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

<a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

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

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];

code = 233; 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]], 10], {i, 1, stages - 1}]

allocated

nonn,easy

Robert Price, May 17 2017

approved

editing

allocated for Robert Price

recycled

allocated

editing

approved

Triangle read by rows: T(n, k) is the number of n-vertex simple graphs with k edges.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 4, 6, 6, 6, 4, 2, 1, 1, 1, 1, 2, 5, 9, 15, 21, 24, 24, 21, 15, 9, 5, 2, 1, 1, 1, 1, 2, 5, 10, 21, 41, 65, 97, 131, 148, 148, 131, 97, 65, 41, 21, 10, 5, 2, 1, 1, 1, 1, 2, 5, 11, 24, 56, 115, 221, 402, 663, 980, 1312, 1557, 1646, 1557, 1312, 980, 663, 402, 221, 115, 56, 24, 11, 5, 2, 1, 1

1,10

Equivalently, nth row gives coefficients of the nth graph polynomial.

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphEdge.html">Graph Edge</a>

Triangle begins:

1;

1, 1;

1, 1, 1, 1;

1, 1, 2, 3, 2, 1, 1;

1, 1, 2, 4, 6, 6, 6, 4, 2, 1, 1;

<<Combinatorica`; CoefficientList[Table[GraphPolynomial[n, x], {n, 10}], x] (* Eric W. Weisstein, May 16 2017 *)

nth row sums to A000088(n).

Sum k*T(n, k) gives A086314(n).

dead

recycled

Eric W. Weisstein, May 16 2017

proposed

editing

editing

proposed