A161342 Number of "ON" cubic cells at n-th stage in simple 3-dimensional cellular automaton: a(n) = A160428(n)/8.

0, 1, 8, 15, 64, 71, 120, 169, 512, 519, 568, 617, 960, 1009, 1352, 1695, 4096, 4103, 4152, 4201, 4544, 4593, 4936, 5279, 7680, 7729, 8072, 8415, 10816, 11159, 13560, 15961, 32768, 32775, 32824, 32873, 33216, 33265, 33608, 33951, 36352, 36401, 36744, 37087, 39488

Offset: 0

Author

Omar E. Pol, Jun 14 2009

Keywords

• nonn

Comments

First differences are in A161343. - Omar E. Pol, May 03 2015

From Gary W. Adamson, Aug 30 2016: (Start)

Let M =

1, 0, 0, 0, 0, ...

8, 0, 0, 0, 0, ...

7, 1, 0, 0, 0, ...

0, 8, 0, 0, 0, ...

0, 7, 1, 0, 0, ...

0, 0, 8, 0, 0, ...

0, 0, 7, 1, 0, ...

...

Then M^k converges to a single nonzero column giving the sequence.

The sequence with offset 1 divided by its aerated variant is (1, 8, 7, 0, 0, 0, ...). (End)

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..16383
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A160410, A160428, A161343, A006046, A130665, A116520, A130667, A116522, A116526, A116525, A360189.

Programs

Maple

b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
a:= n-> subs(x=7, b(n-1)):
seq(a(n), n=0..44);  # Alois P. Heinz, Mar 06 2023

Mathematica

A161342list[nmax_]:=Join[{0},Accumulate[7^DigitCount[Range[0,nmax-1],2,1]]];A161342list[100] (* Paolo Xausa, Aug 05 2023 *)

Formula

From Nathaniel Johnston, Nov 13 2010: (Start)

a(n) = Sum_{k=0..n-1} 7^A000120(k).

a(n) = 1 + 7 * Sum_{k=1..n-1} A151785(k), for n >= 1.

a(2^n) = 2^(3n).

(End)

a(n) = Sum_{k=0..floor(log_2(n))} 7^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Extensions

More terms from Nathaniel Johnston, Nov 13 2010