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 A164032 Number of "ON" cells in a certain 2-dimensional cellular automaton. 1
 1, 9, 4, 36, 4, 36, 16, 144, 4, 36, 16, 144, 16, 144, 64, 576, 4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304, 4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304, 16, 144, 64, 576, 64, 576, 256, 2304, 64, 576, 256, 2304, 256 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This automaton starts with one ON cell and evolves according to the rule that a cell is ON in a given generation if and only if the number of ON cells, among the cell itself and its eight nearest neighbors, was exactly one in the preceding generation. LINKS David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.], which is also available at arXiv:1004.3036v2 N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS FORMULA It appears that this is the self-generating sequence defined by the following process: start with s={1,9} and repeatedly extend by concatenating s with 4*s, thus obtaining {1,9} -> {1,9,4,36} -> {1,9,4,36,4,36,16,144},... , etc. Also, it appears that if n=2^k+j, with n>2 and 1<=j<=2^k, then a(n)=4a(j), with a(1)=1, a(2)=9. From N. J. A. Sloane, Jul 21 2014: (Start) Both of these assertions are not difficult to prove. At generation G = 2^k (k>=1) the ON cells are bounded by a box of edge 2G-1, and in that box there are (G/2)^2 3X3 blocks each containing 9 ON cells (separated by rows of OFF cells of width 1), so a total of a(2^k) = 9*2^(2k-2) ON cells (cf. A002063). This box is full (more precisely, every cell in it has more than one ON neighbor), and at generation G+1 we have just 4 ON cells which are now at the corners of a box of edge 2G+1. Until the next power of 2 there is no interaction between the configurations that grow at the four corners, and so a(2^k+j) = 4a(j), as conjectured. In fact this implies an explicit formula for a(n): a(n) = c*4^wt(floor((n-1)/2)), where c=1 if n is odd, c=9 if n is even, and wt(i) = A000120(i) is the binary weight function. For example, if n=20, [(n-1)/2]=9 which has weight 2, so a(20) = 9*4^2 = 144. (End) EXAMPLE Can be arranged into blocks of length 2^k: 1, 9, 4, 36, 4, 36, 16, 144, 4, 36, 16, 144, 16, 144, 64, 576, 4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304, 4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304, 16, 144, 64, 576, 64, 576, 256, 2304, 64, 576, 256, 2304, 256, ... ... MATHEMATICA wt[i_] := DigitCount[i, 2, 1]; a[n_] := If[OddQ[n], 1, 9] 4^wt[Floor[(n-1)/2]]; Array[a, 61] (* Jean-François Alcover, Oct 08 2018, after N. J. A. Sloane *) PROG a(n) = 4^hammingweight((n-1)\2) * if(n%2, 1, 9); \\ Michel Marcus, Oct 08 2018 CROSSREFS Cf. A000120, A048883, A079315, A122108, A160239, A002063 (last entry in each block) Sequence in context: A014717 A104728 A058093 * A122846 A248309 A317900 Adjacent sequences:  A164029 A164030 A164031 * A164033 A164034 A164035 KEYWORD nonn AUTHOR John W. Layman, Aug 08 2009 STATUS approved

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Last modified May 17 19:26 EDT 2021. Contains 343988 sequences. (Running on oeis4.)