A269713 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 20", based on the 5-celled von Neumann neighborhood.

1, 5, 13, 25, 45, 69, 113, 141, 193, 249, 353, 409, 513, 625, 837, 897, 1013, 1133, 1365, 1485, 1717, 1957, 2421, 2541, 2773, 3013, 3477, 3717, 4181, 4661, 5593, 5717, 5961, 6209, 6697, 6945, 7433, 7929, 8905, 9153, 9641, 10137, 11113, 11609, 12585, 13577

Offset: 0

Author

Robert Price, Mar 04 2016

Keywords

• nonn
• easy

Comments

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

Rules 28, 52, 60, 148, 156, 180, 188, 532, 540, 564, 572, 660, 668, 692 and 700 also generate this sequence.

References

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

Links

• Robert Price, Table of n, a(n) for n = 0..300
• 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
• Index entries for sequences related to cellular automata
• Index to 2D 5-Neighbor Cellular Automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A269711.

Programs

Mathematica

rule=20; stages=300;
ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)