A186410 Number of "ON" cells at n-th stage of three-dimensional version of the cellular automaton A183060 using cubes.

0, 1, 6, 11, 32, 37, 58, 79, 180, 185, 206, 227, 328, 349, 450, 551, 1052, 1057, 1078, 1099, 1200, 1221, 1322, 1423, 1924, 1945, 2046, 2147, 2648, 2749, 3250, 3751, 6252, 6257, 6278, 6299, 6400, 6421, 6522, 6623

Offset: 0

Author

Omar E. Pol, Feb 21 2011

Keywords

• nonn

Comments

The sequence gives the total number of cells turned ON after n stages in a cellular automaton based on Z^3 lattice in the same way that A183060 is based on the Z^2 lattice. In general here each cell has six neighbors.

It appears that after 2^k stages the structure resembles a pyramid. For the first differences see A186411.

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, pp. 32-33.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A139250, A147562, A151781, A183060.

Programs

Mathematica

a[n_] := n + (4/5) Sum[5^DigitCount[i, 2, 1], {i, n - 1}]; Array[a, 40, 0] (* Michael De Vlieger, Nov 02 2022 *)

Formula

From Nathaniel Johnston, Mar 14 2011: (Start)

a(n) = n + (4/5)*(Sum_{i=1..n-1} 5^A000120(i)).

a(2^n) = 2^n + (4/5)*(6^n - 1).

(End)

Extensions

More terms from Nathaniel Johnston, Mar 14 2011