A186410 - OEIS

%I #22 Nov 02 2022 11:53:53

%S 0,1,6,11,32,37,58,79,180,185,206,227,328,349,450,551,1052,1057,1078,

%T 1099,1200,1221,1322,1423,1924,1945,2046,2147,2648,2749,3250,3751,

%U 6252,6257,6278,6299,6400,6421,6522,6623

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

%C 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.

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

%H Michael De Vlieger, <a href="/A186410/b186410.txt">Table of n, a(n) for n = 0..1000</a>

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]

%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, pp. 32-33.

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F From _Nathaniel Johnston_, Mar 14 2011: (Start)

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

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

%F (End)

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

%Y Cf. A139250, A147562, A151781, A183060.

%K nonn

%O 0,3

%A _Omar E. Pol_, Feb 21 2011

%E More terms from _Nathaniel Johnston_, Mar 14 2011