A161342 - OEIS

%I #34 Aug 05 2023 13:18:07

%S 0,1,8,15,64,71,120,169,512,519,568,617,960,1009,1352,1695,4096,4103,

%T 4152,4201,4544,4593,4936,5279,7680,7729,8072,8415,10816,11159,13560,

%U 15961,32768,32775,32824,32873,33216,33265,33608,33951,36352,36401,36744,37087,39488

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

%C First differences are in A161343. - _Omar E. Pol_, May 03 2015

%C From _Gary W. Adamson_, Aug 30 2016: (Start)

%C Let M =

%C   1, 0, 0, 0, 0, ...

%C   8, 0, 0, 0, 0, ...

%C   7, 1, 0, 0, 0, ...

%C   0, 8, 0, 0, 0, ...

%C   0, 7, 1, 0, 0, ...

%C   0, 0, 8, 0, 0, ...

%C   0, 0, 7, 1, 0, ...

%C   ...

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

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

%H Alois P. Heinz, <a href="/A161342/b161342.txt">Table of n, a(n) for n = 0..16383</a>

%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_, Nov 13 2010: (Start)

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

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

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

%F (End)

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

%p b:= proc(n) option remember; `if`(n<0, 0,

%p       b(n-1)+x^add(i, i=Bits[Split](n)))

%p     end:

%p a:= n-> subs(x=7, b(n-1)):

%p seq(a(n), n=0..44);  # _Alois P. Heinz_, Mar 06 2023

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

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

%K nonn

%O 0,3

%A _Omar E. Pol_, Jun 14 2009

%E More terms from _Nathaniel Johnston_, Nov 13 2010