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 A160412 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition). 8
 0, 3, 12, 21, 48, 57, 84, 111, 192, 201, 228, 255, 336, 363, 444, 525, 768, 777, 804, 831, 912, 939, 1020, 1101, 1344, 1371, 1452, 1533, 1776, 1857, 2100, 2343, 3072, 3081, 3108, 3135, 3216, 3243, 3324, 3405, 3648, 3675, 3756, 3837, 4080, 4161, 4404, 4647 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Omar E. Pol, Nov 10 2009: (Start) On the infinite square grid, consider the outside corner of an infinite square. We start at round 0 with all cells in the OFF state. The rule: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells. At round 1, we turn ON three cells around the corner of the infinite square, forming a concave-convex hexagon with three exposed vertices. At round 2, we turn ON nine cells around the hexagon. At round 3, we turn ON nine other cells. Three cells around of every corner of the hexagon. And so on. Shows a fractal-like behavior similar to the toothpick sequence A153006. For the first differences see the entry A162349. For more information see A160410, which is the main entry for this sequence. (End) LINKS Michael De Vlieger, Table of n, a(n) for n = 0..1000 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.] Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31. Omar E. Pol, Illustration of initial terms [From Omar E. Pol, Nov 10 2009] N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS Index entries for sequences related to cellular automata - Omar E. Pol, Nov 10 2009 FORMULA From Omar E. Pol, Nov 10 2009: (Start) a(n) = A160410(n)*3/4. a(0) = 0, a(n) = A130665(n-1)*3, for n>0. (End) EXAMPLE If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below: ...77..77..77..77 ...766667..766667 ....6556....6556. ....654444444456. ...76643344334667 ...77.43222234.77 ......44211244... 00000000001244... 00000000002234.77 00000000004334667 0000000000444456. 0000000000..6556. 0000000000.766667 0000000000.77..77 0000000000....... 0000000000....... 0000000000....... MATHEMATICA a[n_] := 3*Sum[3^DigitCount[k, 2, 1], {k, 0, n - 1}]; Array[a, 48, 0] (* Michael De Vlieger, Nov 01 2022 *) CROSSREFS Cf. A139250, A139251, A153006, A152980, A160410, A160414. Cf. A130665, A162349. - Omar E. Pol, Nov 10 2009 Sequence in context: A044436 A210282 A160167 * A091846 A061262 A051656 Adjacent sequences: A160409 A160410 A160411 * A160413 A160414 A160415 KEYWORD nonn AUTHOR Omar E. Pol, May 20 2009, Jun 01 2009 EXTENSIONS More terms from Omar E. Pol, Nov 10 2009 Edited by Omar E. Pol, Nov 11 2009 More terms from Nathaniel Johnston, Nov 06 2010 More terms from Colin Barker, Apr 19 2015 STATUS approved

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