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A169707 Total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighborhood. 26
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

REFERENCES

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.

LINKS

N. J. A. Sloane and Vincenzo Librandi, Table of n, a(n) for n = 1..513, computed from the definition, not using the conjectured formula. [The first 200 terms were found by Vincenzo Librandi]

David Applegate, The movie version

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), pp. 157-191. arXiv:1004.3036 [math.CO].

N. J. A. Sloane, Table of n, a(n) for n=1..8192 assuming the recurrence is correct. Note that these terms are merely conjectures.

N. J. A. Sloane, Illustration of first 24 generations

N. J. A. Sloane, Illustration of first 24 generations of the NW-NE-SE-SW version of this CA

N. J. A. Sloane, Illustration of generation 7 of the NW-NE-SE-SW version of this CA (containing a(7) = 57 ON cells)

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to cellular automata

FORMULA

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.

a(n) = 1 + 2*(A139250(n) - A160552(n)) = A160164(n) - A170903(n) = A187220(n) + 2*(A160552(n-1)). - Omar E. Pol, Feb 18 2015

It appears that a(n) = A162795(n) = A147562(n), if n is a member of A048645, otherwise a(n) > A162795(n) > A147562(n). - Omar E. Pol, Feb 19 2015

It appears that a(n) = 1 + 4*A255747(n-1). - Omar E. Pol, Mar 05 2015

It appears that a(n) = 1 + 4*(A139250(n-1) - (a(n-1) - 1)/4), n > 1. - Omar E. Pol, Jul 24 2015

It appears that a(2n) = 1 + 4*A162795(n). - Omar E. Pol, Jul 04 2017

EXAMPLE

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, ...

From Omar E. Pol, Feb 18 2015: (Start)

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)

MAPLE

(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;

# a loop that produces M steps of A169707 and A169708:

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:

MATHEMATICA

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]

CROSSREFS

Cf. A169708 (first differences), A147562, A147582, A169648, A169649, A169709, A169710, A246333, A246334, A246335, A246336, A253098 (partial sums).

Cf. also A162795, A150552, A160104, A170903, A048645, A187200.

See A253088 for the analogous CA using Rule 750 and a 9-celled neighborhood.

Sequence in context: A162795 A255366 A269522 * A256260 A256250 A256138

Adjacent sequences:  A169704 A169705 A169706 * A169708 A169709 A169710

KEYWORD

nonn,look

AUTHOR

N. J. A. Sloane, Apr 17 2010

EXTENSIONS

Edited (added formula, illustration, etc.) by N. J. A. Sloane, Aug 30 2014

Offset changed to 1 by N. J. A. Sloane, Feb 09 2015

STATUS

approved

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Last modified February 19 18:31 EST 2018. Contains 299356 sequences. (Running on oeis4.)