A275667 -id:A275667 - cp's OEIS frontend

A170842 G.f.: Product_{k>=1} (1 + 2x^(2^k-1) + 3x^(2^k)).

1, 2, 3, 2, 7, 12, 9, 2, 7, 12, 13, 20, 45, 54, 27, 2, 7, 12, 13, 20, 45, 54, 31, 20, 45, 62, 79, 150, 243, 216, 81, 2, 7, 12, 13, 20, 45, 54, 31, 20, 45, 62, 79, 150, 243, 216, 85, 20, 45, 62, 79, 150, 243, 224, 133, 150, 259, 344, 537, 936, 1161, 810, 243, 2, 7, 12, 13, 20, 45

Offset: 0

N. J. A. Sloane, Jan 02 2010

From Omar E. Pol, Apr 10 2021: (Start)
It appears that this is also an irregular triangle read by rows (see the example).
It appears that right border gives A000244.
It appears that row sums give A052934. (End)

			From _Omar E. Pol_, Apr 10 2021: (Start)
Written as an irregular triangle in which row lengths are A000079 the sequence begins:
1;
2, 3;
2, 7, 12, 9;
2, 7, 12, 13, 20, 45, 54, 27;
2, 7, 12, 13, 20, 45, 54, 31, 20, 45, 62, 79, 150, 243, 216, 81;
2, 7, 12, 13, 20, 45, 54, 31, 20, 45, 62, 79, 150, 243, 216, 85, 20, 45, 62, ...
(End)

John Tyler Rascoe, Table of n, a(n) for n = 0..8192
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

Cf. A000079, A000244, A052934, A275667.

CoefficientList[Series[Product[1+2x^(2^k-1)+3x^2^k,{k,10}],{x,0,70}],x] (* Harvey P. Dale, Apr 09 2021 *)

D_x(N) = {my( x='x+O('x^N));Vec(prod(k=1,logint(N,2)+1,(1+2*x^(2^k-1)+3*x^(2^k))))}
D_x((2^6)+1) \\ John Tyler Rascoe, Aug 16 2024

A383369 Population of elementary triangular automaton rule 90 at generation n, starting from a lone 1 cell at generation 0.

Original entry on oeis.org

1, 4, 6, 12, 6, 24, 24, 48, 6, 24, 36, 72, 24, 96, 96, 192, 6, 24, 36, 72, 36, 144, 144, 288, 24, 96, 144, 288, 96, 384, 384, 768, 6, 24, 36, 72, 36, 144, 144, 288, 36, 144, 216, 432, 144, 576, 576, 1152, 24, 96, 144, 288, 144, 576, 576, 1152, 96, 384, 576, 1152, 384, 1536, 1536, 3072, 6

Offset: 0

Paul Cousin, Apr 24 2025

nonn

An Elementary Triangular Automaton (ETA) is a cellular automaton in the triangular grid where cells hold binary states and rules are local to the first neighborhood. There are 256 possible ETA rules.
Rule 90 (1011010 in binary):
  -----------------------------------------------
  |state of the cell            |1|1|1|1|0|0|0|0|
  |sum of the neighbors' states |3|2|1|0|3|2|1|0|
  |cell's next state            |0|1|0|1|1|0|1|0|
  -----------------------------------------------
This is one of the 4 ETA rules (85, 90, 165 and 170) that replicates the pattern given as initial condition.

			Written as an irregular triangle with row lengths A000079, starting from n=1, the sequence begins:
  4;
  6, 12;
  6, 24, 24, 48;
  6, 24, 36, 72, 24, 96, 96, 192;
  6, 24, 36, 72, 36, 144, 144, 288, 24, 96, 144, 288, 96, 384, 384, 768;
...
It appears that the right border gives A110594.

Paul Cousin, Table of n, a(n) for n = 0..16384
Paul Cousin, Illustration for n = 0..128
Paul Cousin, Triangular Automata
Paul Cousin, Rule 90
Paul Cousin, Triangular Automata Integer Sequences
Paul Cousin, Triangular Automata: The 256 Elementary Cellular Automata of the Two-Dimensional Plane, Complex Systems, 33(3), 2024, pp. 253-276.

Pattern replicating ETA rules: A275667 (rule 170).
A247640 is a bisection.
A246035 is the analog on the square cells.

cp's OEIS Frontend

A170842 G.f.: Product_{k>=1} (1 + 2x^(2^k-1) + 3x^(2^k)).

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