A186410 -id:A186410 - cp's OEIS frontend

A186411 First differences of A186410.

0, 1, 5, 5, 21, 5, 21, 21, 101, 5, 21, 21, 101, 21, 101, 101, 501, 5, 21, 21, 101, 21, 101, 101, 501, 21, 101, 101, 501, 101, 501, 501, 2501, 5, 21, 21, 101, 21, 101, 101, 501, 21, 101, 101, 501, 101, 501, 501, 2501, 21, 101, 101, 501, 101, 501, 501, 2501, 101

Offset: 0

Omar E. Pol, Feb 21 2011

nonn

Number of cells turned "ON" at h-th stage of the three-dimensional cellular automaton of A186410. In other words: number of cubes added at n-th stage to the structure of A186410.

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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
Index entries for sequences related to cellular automata.

Cf. A000120, A147582, A151779, A186410.

a[n_] := 1 + 4*5^(DigitCount[n-1, 2, 1]-1); a[0] = 0; a[1] = 1; Array[a, 100, 0] (* Amiram Eldar, Aug 01 2023 *)

a(n) = 1 + 4*5^(A000120(n-1)-1), n >= 2. - Nathaniel Johnston, Mar 22 2011

More terms from Nathaniel Johnston, Mar 22 2011

More terms from Amiram Eldar, Aug 01 2023

A183060 Number of "ON" cells at n-th stage in a simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 4, 7, 14, 17, 24, 31, 50, 53, 60, 67, 86, 93, 112, 131, 186, 189, 196, 203, 222, 229, 248, 267, 322, 329, 348, 367, 422, 441, 496, 551, 714, 717, 724, 731, 750, 757, 776, 795, 850, 857, 876, 895, 950, 969, 1024, 1079, 1242, 1249, 1268, 1287

Offset: 0

Omar E. Pol, Feb 20 2011

nonn

On the semi-infinite square grid, start with all cells OFF.
Turn a single cell to the ON state in row 1.
At each subsequent step, each cell with exactly one neighbor ON is turned ON, and everything that is already ON remains ON.
The sequence gives the number of "ON" cells after n stages. A183061 (the first differences) gives the number of cells turned "ON" at the n-th stage.
Note that this is just half plus the rest of the center line of the cellular automaton described in A147562.
After 2^k stages the structure resembles an isosceles right triangle. For a three-dimensional version using cubes see A186410. For more information see A147562.

			Illustration of the structure after eight stages in which we label the generations of cells turned ON by consecutive numbers:
         8
        878
       8 6 8
      8765678
     8 8 4 8 8
    878 434 878
   8 6 4 2 4 6 8
  876543212345678
...................
There are 50 "ON" cells so a(8) = 50.

JungHwan Min, Table of n, a(n) for n = 0..2500
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, pp. 31-32.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A139250, A147562, A147582, A147610, A151920, A183061, A186410.

A183060[0] = 0; A183060[n_] := Total[With[{m = n - 1}, CellularAutomaton[{4042387958, 2, {{0, 1}, {-1, 0}, {0, 0}, {1, 0}, {0, -1}}}, {{{1}}, 0}, {{{m}}, -m}]], 2] (* JungHwan Min, Jan 24 2016 *)
A183060[0] = 0; A183060[n_] := Total[With[{m = n - 1}, CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, {{{m}}, -m}]], 2] (* JungHwan Min, Jan 24 2016 *)

a(n) = n + (A147562(n) - 1)/2, n >= 1.
a(n) = n + 2*A151920(n-2), n >= 2.
a(2^n) = A076024(n+1). - Nathaniel Johnston, Mar 14 2011

Comments edited by Omar E. Pol, Mar 19 2011 at the suggestion of John W. Layman and Franklin T. Adams-Watters

cp's OEIS Frontend

A186411 First differences of A186410.

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