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A169707
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Total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighborhood.
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26
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1, 5, 9, 21, 25, 37, 57, 85, 89, 101, 121, 149, 169, 213, 281, 341, 345, 357, 377, 405, 425, 469, 537, 597, 617, 661, 729, 805, 889, 1045, 1241, 1365, 1369, 1381, 1401, 1429, 1449, 1493, 1561, 1621, 1641, 1685, 1753, 1829, 1913, 2069, 2265, 2389, 2409, 2453, 2521
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OFFSET
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1,2
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COMMENTS
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Square grid, 4 neighbors per cell (N, E, S, W cells), turn ON iff exactly 1 or 3 neighbors are ON; once ON, cells stay ON.
The terms agree with those of A246335 for n <= 11, although the configurations are different starting at n = 7. - N. J. A. Sloane, Sep 21 2014
Offset 1 is best for giving a formula for a(n), although the Maple and Mathematica programs index the states starting at state 0.
It appears that this shares infinitely many terms with both A162795 and A147562, see Formula section and Example section. - Omar E. Pol, Feb 19 2015
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.
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LINKS
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FORMULA
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a(2^k + i) = (4^(k+1)-1)/3 + 4*A246336(i), for k >= 0, 0 <= i < 2^k. For example, if n = 15 = 2^3 + 7, so k=3, i=7, we have a(15) = (4^4-1)/3 + 4*A246336(7) = 85 + 4*49 = 281.
It appears that a(n) = 1 + 4*(A139250(n-1) - (a(n-1) - 1)/4), n > 1. - Omar E. Pol, Jul 24 2015
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EXAMPLE
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Divides naturally into blocks of sizes 1,2,4,8,16,...:
1,
5, 9,
21, 25, 37, 57,
85, 89, 101, 121, 149, 169, 213, 281, <- terms 8 through 15
341, 345, 357, 377, 405, 425, 469, 537, 597, 617, 661, 729, 805, 889, 1045, 1241,
1365, 1369, 1381, 1401, 1429, 1449, 1493, 1561, 1621, 1641, 1685, 1753, 1829, 1913, 2069, 2265, 2389, 2409, 2453, 2521, ...
Also, written as an irregular triangle T(j,k), k>=1, in which the row lengths are the terms of A011782:
1;
5;
9, 21;
25, 37, 57, 85;
89, 101, 121, 149, 169, 213, 281, 341;
345, 357, 377, 405, 425, 469, 537, 597, 617, 661, 729, 805, 889, 1045, 1241, 1365;
The right border gives the positive terms of A002450.
It appears that T(j,k) = A162795(j,k) = A147562(j,k), if k is a power of 2, for example: it appears that the three mentioned triangles only share the elements from the columns 1, 2, 4, 8, 16, ...
(End)
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MAPLE
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(Maple program that uses the actual definition of the automaton, rather than the (conjectured) formula, from N. J. A. Sloane, Feb 15 2015):
# Count terms in a polynomial:
C := f->`if`(type(f, `+`), nops(f), 1);
# Replace all nonzero coeffts by 1:
bool := proc(f) local ix, iy, f2, i, t1, t2, A;
f2:=expand(f);
if whattype(f) = `+` then
t1:=nops(f2); A:=0;
for i from 1 to t1 do t2:=op(i, f2); ix:=degree(t2, x); iy:=degree(t2, y);
A:=A+x^ix*y^iy; od: A;
else ix:=degree(f2, x); iy:=degree(f2, y); x^ix*y^iy;
fi;
end;
M:=20;
F:=x*y+x/y+1/x*y+1/x/y mod 2;
GG[0]:=1;
for n from 1 to M do dd[n]:=expand(F*GG[n-1]) mod 2;
GG[n]:=bool(GG[n-1]+dd[n]);
lprint(n, C(GG[n]), C(GG[n]-GG[n-1])); od:
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MATHEMATICA
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Map[Function[Apply[Plus, Flatten[ #1]]], CellularAutomaton[{ 750, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 100]]
ArrayPlot /@ CellularAutomaton[{750, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 23]
(* The next two lines deal with the equivalent CA based on neighbors NW, NE, SE, SW. This is to facilitate the comparison with A246333 and A246335 *)
Map[Function[Apply[Plus, Flatten[ #1]]], CellularAutomaton[{ 750, {2, {{2, 0, 2}, {0, 1, 0}, {2, 0, 2}}}, {1, 1}}, {{{1}}, 0}, 100]]
ArrayPlot /@ CellularAutomaton[{750, {2, {{2, 0, 2}, {0, 1, 0}, {2, 0, 2}}}, {1, 1}}, {{{1}}, 0}, 23]
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CROSSREFS
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Cf. A169708 (first differences), A147562, A147582, A169648, A169649, A169709, A169710, A246333, A246334, A246335, A246336, A253098 (partial sums).
See A253088 for the analogous CA using Rule 750 and a 9-celled neighborhood.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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