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A160119 A three-dimensional version of the cellular automaton A160118, using cubes. 6

%I #25 Aug 01 2023 06:55:51

%S 0,1,27,35,235,243,443,499,1899,1907,2107,2163,3563,3619,5019,5411,

%T 15211,15219,15419,15475,16875,16931,18331,18723,28523,28579,29979,

%U 30371,40171,40563,50363,53107,121707,121715,121915,121971,123371,123427,124827,125219,135019

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

%C Each cell has 26 neighbors.

%C Differs from A160379 in the same way that A160118 differs from A160117. - _N. J. A. Sloane_, Jan 01 2010

%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 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 24 2011: (Start)

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

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

%F 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)):

%F 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.

%F 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)

%t 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 *)

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

%K nonn

%O 0,3

%A _Omar E. Pol_, May 05 2009

%E More terms from _Omar E. Pol_, May 11 2009

%E Edited by _N. J. A. Sloane_, Sep 05 2009

%E a(8)-a(32) from _Nathaniel Johnston_, Mar 24 2011

%E More terms from _Amiram Eldar_, Aug 01 2023

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Last modified March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)