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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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9
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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.
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LINKS
| David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
David Applegate, The movie version [From Omar E. Pol (info(AT)polprimos.com), Oct 10 2010]
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EXAMPLE
| 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
| 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: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 14 2009]
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CROSSREFS
| Cf. A139250, A160117, A160118, A160721, A160740, A147562, A160410, A160414, A160412, A160416, A160796.
Sequence in context: A087940 A092387 A175364 * A147552 A147562 A162795
Adjacent sequences: A160717 A160718 A160719 * A160721 A160722 A160723
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KEYWORD
| nonn
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AUTHOR
| Omar E. Pol (info(AT)polprimos.com), May 25 2009, May 29 2009, Jun 28 2009
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EXTENSIONS
| Edited by N. J. A. Sloane, Jun 26 2009
More terms from David Applegate (david(AT)research.att.com), Jul 03 2009
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