|
|
A169699
|
|
Total number of ON cells at stage n of two-dimensional 5-neighbor outer totalistic cellular automaton defined by "Rule 510".
|
|
17
|
|
|
1, 5, 12, 25, 28, 56, 56, 113, 60, 120, 120, 240, 120, 240, 240, 481, 124, 248, 248, 496, 248, 496, 496, 992, 248, 496, 496, 992, 496, 992, 992, 1985, 252, 504, 504, 1008, 504, 1008, 1008, 2016, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032, 504, 1008, 1008, 2016
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
We work on the square grid. Each cell has 4 neighbors, N, S, E, W. If none of your 4 neighbors are ON, your state does not change. If all 4 of your neighbors are ON, your state flips. In all other cases you turn ON. We start with one ON cell.
As observed by Packard and Wolfram (see Fig. 2), a slice along the E-W line shows the successive states of the 1-D CA Rule 126 (see A071035, A071051).
|
|
REFERENCES
|
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
|
|
LINKS
|
|
|
FORMULA
|
For n>0, it is easy to show that if 2^k <= n < 2^(k+1) then a(n) =
(2^(k+1)-1)*2^(1+wt(n)), where wt is the binary weight A000120, except that if n is a power of 2 we must add 1 to the result.
|
|
EXAMPLE
|
When arranged into blocks of sizes 1,1,2,4,8,16,...:
1,
5,
12, 25,
28, 56, 56, 113,
60, 120, 120, 240, 120, 240, 240, 481,
124, 248, 248, 496, 248, 496, 496, 992, 248, 496, 496, 992, 496, 992, 992, 1985,
252, 504, 504, 1008, 504, 1008, 1008, 2016, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032,
..., the initial terms in the rows (after the initial rows) have the form 2^m-4 and the final terms are given by A092440. The row beginning with 2^m-4 is divisible by 2^(m-2)-1 (see formula).
|
|
MAPLE
|
A000120 := proc(n) add(i, i=convert(n, base, 2)) end:
ht:=n->floor(log[2](n));
f:=proc(n) local a, t1;
if n=0 then 1 else
a:=(2^(ht(n)+1)-1)*2^(1+A000120(n));
if 2^log[2](n)=n then a:=a+1; fi; a; fi; end;
|
|
MATHEMATICA
|
Map[Function[Apply[Plus, Flatten[ #1]]], CellularAutomaton[{ 510, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 100]]
ArrayPlot /@ CellularAutomaton[{510, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 28]
|
|
CROSSREFS
|
See A253089 for 9-celled neighborhood version.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Entry revised with more precise definition, formula and additional information, N. J. A. Sloane, Aug 24 2014
|
|
STATUS
|
approved
|
|
|
|