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A160720
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Number of "ON" cells at n-th stage in 2-dimensional cellular automaton (see Comments for precise definition).
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17
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0, 1, 5, 9, 21, 25, 37, 49, 77, 81, 93, 105, 133, 145, 173, 201, 261, 265, 277, 289, 317, 329, 357, 385, 445, 457, 485, 513, 573, 601, 661, 721, 845, 849, 861, 873, 901, 913, 941, 969, 1029, 1041, 1069, 1097, 1157, 1185, 1245, 1305, 1429, 1441, 1469, 1497
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OFFSET
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0,3
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COMMENTS
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We work on the vertices of the square grid Z^2, and define the neighbors of a cell to be the four closest cells along the diagonals.
We start at stage 0 with all cells in OFF state.
At stage 1, we turn ON a single cell at the origin.
Once a cell is ON it stays ON.
At each subsequent stage, a cell in turned ON if exactly one of its neighboring cells that are no further from the origin is ON.
The "no further from the origin" condition matters for the first time at stage 8, when only A160721(8) = 28 cells are turned ON, and a(8) = 77. In contrast, A147562(8) = 85, A147582(8) = 36.
This CA also arises as the cross-section in the (X,Y)-plane of the CA in A151776.
In other words, a cell is turned ON if exactly one of its vertices touches an exposed vertex of a ON cell of the previous generation. A special rule for this sequence is that every ON cell has only one vertex that should be considered not exposed: its nearest vertex to the center of the structure.
Analog to the "outward" version (A266532) of the Y-toothpick cellular automaton of A160120 on the triangular grid, but here we have ON cells on the square grid. See also the formula section. - Omar E. Pol, Jan 19 2016
This cellular automaton can be interpreted as the outward version of the Ulam-Warburton two-dimensional cellular automaton (see A147562). - Omar E. Pol, Jun 22 2017
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LINKS
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FORMULA
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EXAMPLE
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If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...............9
.8.8.8.8.8.8.8.8.
..7...7...7...7..
.8.6.6.....6.6.8.
....5.......5....
.8.6.4.4.4.4.6.8.
..7...3...3...7..
.8...4.2.2.4...8.
........1........
.8...4.2.2.4...8.
..7...3...3...7..
.8.6.4.4.4.4.6.8.
....5.......5....
.8.6.6.....6.6.8.
..7...7...7...7..
.8.8.8.8.8.8.8.8.
9...............9
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MAPLE
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cellOn := [[0, 0]] : bbox := [0, 0, 0, 0]: # llx, lly, urx, ury isOn := proc(x, y, L) local i ; for i in L do if op(1, i) = x and op(2, i) = y then RETURN(true) ; fi; od: RETURN(false) ; end: bb := proc(L) local mamin, i; mamin := [0, 0, 0, 0] ; for i in L do mamin := subsop(1=min(op(1, mamin), op(1, i)), mamin) ; mamin := subsop(2=min(op(2, mamin), op(2, i)), mamin) ; mamin := subsop(3=max(op(1, mamin), op(1, i)), mamin) ; mamin := subsop(4=max(op(2, mamin), op(2, i)), mamin) ; od: mamin ; end: for gen from 2 to 80 do nGen := [] ; print(nops(cellOn)) ; for x from op(1, bbox)-1 to op(3, bbox)+1 do for y from op(2, bbox)-1 to op(4, bbox)+1 do # not yet in list? if not isOn(x, y, cellOn) then
# loop over 4 neighbors of (x, y) non := 0 ; for dx from -1 to 1 by 2 do for dy from -1 to 1 by 2 do # test of a neighbor nearer to origin if x^2+y^2 >= (x+dx)^2+(y+dy)^2 then if isOn(x+dx, y+dy, cellOn) then non := non+1 ; fi; fi; od: od: # exactly one neighbor on: add to nGen if non = 1 then nGen := [op(nGen), [x, y]] ; fi; fi; od: od: # merge old and new generation cellOn := [op(cellOn), op(nGen)] ; bbox := bb(cellOn) ; od: # R. J. Mathar, Jul 14 2009
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MATHEMATICA
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A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (Function[pos, CellularAutomaton[{FromDigits[Boole[#[[2, 2]] == 1 || Count[Flatten[#], 1] == 1 && Count[Extract[#, pos], 1] == 1] & /@ Tuples[{1, 0}, {3, 3}], 2], 2, {1, 1}}, {{{1}}, 0}, {{{m}}, {-m, m}, {-m, m}}]] /@ Partition[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}}, 2, 1, 1])], 2] (* JungHwan Min, Jan 23 2016 *)
A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (CellularAutomaton[{#, 2, {1, 1}}, {{{1}}, 0}, {{{m}}, {-m, m}, {-m, m}}] & /@ {13407603346151304507647333602124270744930157291580986197148043437687863763597662002711256755796972443613438635551055889478487182262900810351549134401372178, 13407603346151304507647333602124270744930157291580986197148043437687863763597777794800494071992396014598447323458909159463152822826940267935557047531012112, 13407603346151304507647333602124270744930157291580986197148043437687863763597777794800494071992396014598447323458909159463152822826940286382301121240563712, 13407603346151304507647333602124270744930157291580986197148043437687863763597662002711256755796972443613438635551055889478487182262900828798293208110923778})], 2] (* JungHwan Min, Jan 23 2016 *)
A160720[0]=0; A160720[n_]:=Total[With[{m = n - 1}, BitOr @@ (CellularAutomaton[{46, {2, ReplacePart[ArrayPad[{{1}}, 1], # -> 2]}, {1, 1}}, {{{1}}, 0}, {{{m}}, All, All}] & /@ Partition[{{-1, -1}, {-1, 1}, {1, 1}, {1, -1}}, 2, 1, 1])], 2] (* JungHwan Min, Jan 24 2016 *)
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CROSSREFS
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Cf. A139250, A147562, A159912, A160117, A160118, A160120, A160721, A160740, A147562, A160410, A160414, A160412, A160416, A160796, A266532, A267700.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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