Revision History for A271007
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| A271007
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First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 245", based on the 5-celled von Neumann neighborhood.
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published version)
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#11 by Charles R Greathouse IV at Fri Jul 26 21:16:35 EDT 2024
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| NAME
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First differences of number of active (ON,, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 245", based on the 5-celled von Neumann neighborhood.
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Discussion
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| Fri Jul 26
| 21:16
| OEIS Server: https://oeis.org/edit/global/2997
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#10 by Joerg Arndt at Tue Mar 29 13:20:16 EDT 2016
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#9 by Robert Price at Tue Mar 29 13:08:49 EDT 2016
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#8 by Robert Price at Tue Mar 29 13:08:46 EDT 2016
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| LINKS
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Robert Price, <a href="/A271007/b271007.txt">Table of n, a(n) for n = 0..127</a>
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approved
editing
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#7 by Peter Luschny at Tue Mar 29 03:07:02 EDT 2016
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#6 by Michel Marcus at Tue Mar 29 02:46:38 EDT 2016
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#5 by Colin Barker at Mon Mar 28 13:53:46 EDT 2016
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#4 by Colin Barker at Mon Mar 28 13:53:32 EDT 2016
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| FORMULA
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Conjectures from Colin Barker, Mar 28 2016: (Start)
a(n) = 6+9*(-1)^n+4*n+4*(-1)^n*n^2 for n>5.
a(n) = 4*n^2+4*n+15 for n>5 and even.
a(n) = -4*n^2+4*n-3 for n>5 and odd.
a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>8.
G.f.: (7+3*x+22*x^2+3*x^3+3*x^4-3*x^5-4*x^6-3*x^7+8*x^8-4*x^10) / ((1-x)^2*(1+x)^3).
(End)
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proposed
editing
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#3 by Robert Price at Mon Mar 28 12:22:17 EDT 2016
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#2 by Robert Price at Mon Mar 28 12:22:12 EDT 2016
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allocatedFirst differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 245", based on the 5-celled forvon RobertNeumann Priceneighborhood.
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| DATA
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7, -4, 40, -31, 99, -87, 183, -171, 303, -291, 455, -443, 639, -627, 855, -843, 1103, -1091, 1383, -1371, 1695, -1683, 2039, -2027, 2415, -2403, 2823, -2811, 3263, -3251, 3735, -3723, 4239, -4227, 4775, -4763, 5343, -5331, 5943, -5931, 6575, -6563, 7239
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| OFFSET
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0,1
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| COMMENTS
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Initialized with a single black (ON) cell at stage zero.
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| REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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| LINKS
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N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
<a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
<a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
<a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
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| MATHEMATICA
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CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=245; stages=128;
rule=IntegerDigits[code, 2, 10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]], Range[k+1-n, k-1+n]], {j, k+1-n, k-1+n}], {n, 1, k}];
on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
Table[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *)
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| CROSSREFS
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Cf. A271004.
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| KEYWORD
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allocated
sign,easy
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| AUTHOR
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Robert Price, Mar 28 2016
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| STATUS
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approved
editing
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