A160119 A three-dimensional version of the cellular automaton A160118, using cubes.

0, 1, 27, 35, 235, 243, 443, 499, 1899, 1907, 2107, 2163, 3563, 3619, 5019, 5411, 15211, 15219, 15419, 15475, 16875, 16931, 18331, 18723, 28523, 28579, 29979, 30371, 40171, 40563, 50363, 53107, 121707, 121715, 121915, 121971, 123371, 123427, 124827, 125219, 135019

Offset

0,3

Comments

Each cell has 26 neighbors.

Differs from A160379 in the same way that A160118 differs from A160117. - N. J. A. Sloane, Jan 01 2010

Links

Table of n, a(n) for n=0..40.

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.

Formula

From Nathaniel Johnston, Mar 24 2011: (Start)

a(2n-1) = 27 + 8*Sum_{k=1..n-1}A151785(k) + 200*Sum_{k=1..n-2}A151785(k), n >= 2.

a(2n) = 27 + 8*Sum_{k=1..n-1}A151785(k) + 200*Sum_{k=1..n-1}A151785(k), n >= 1.

In general, a d-dimensional version of the cellular automaton A160118 has its cell count given by the following formulas (where wt(k) = A000120(k)):

a(2n-1) = 3^d + (2^d)*Sum_{k=1..n-1}(2^d-1)^(wt(k)-1) + (2^d)*(3^d-2)*Sum_{k=1..n-2}(2^d-1)^(wt(k)-1), n >= 2.

a(2n) = 3^d + (2^d)*Sum_{k=1..n-1}(2^d-1)^(wt(k)-1) + (2^d)*(3^d-2)*Sum_{k=1..n-1}(2^d-1)^(wt(k)-1), n >= 1. (End)

Mathematica

With[{d = 3}, wt[n_] := DigitCount[n, 2, 1]; a[n_] := If[OddQ[n], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]; a[0] = 0; a[1] = 1; Array[a, 50, 0]] (* Amiram Eldar, Aug 01 2023 *)

Crossrefs

Cf. A000120, A139250, A139251, A151785, A160117, A160118, A160379.

Sequence in context: A226191 A098883 A326893 * A160379 A351473 A152053

Adjacent sequences: A160116 A160117 A160118 * A160120 A160121 A160122

Keyword

nonn

Author

Omar E. Pol, May 05 2009

Extensions

More terms from Omar E. Pol, May 11 2009

Edited by N. J. A. Sloane, Sep 05 2009

a(8)-a(32) from Nathaniel Johnston, Mar 24 2011

More terms from Amiram Eldar, Aug 01 2023

Status

approved