A247640 -id:A247640 - cp's OEIS frontend

A164908 a(n) = (3*4^n - 0^n)/2.

1, 6, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, 25165824, 100663296, 402653184, 1610612736, 6442450944, 25769803776, 103079215104, 412316860416, 1649267441664, 6597069766656, 26388279066624, 105553116266496, 422212465065984, 1688849860263936

Offset: 0

Klaus Brockhaus, Aug 31 2009

Binomial transform of A164907. Inverse binomial transform of A057651.
Partial sums are in A083420.
Decimal representations of the n-th iterations of elementary cellular automata rules 14, 46, 142 and 174 generate this sequence (see A266298 and A266299). - Karl V. Keller, Jr., Aug 31 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata.
Index entries for linear recurrences with constant coefficients, signature (4).

Equals 1 followed by A002023 (6*4^n). Essentially the same as A084509.

Cf. A164907, A057651, A083420 (2*4^n-1), A247640, A266298, A266299.

Magma
```
[ (3*4^n-0^n)/2: n in [0..22] ];
```

a[n_]:=(MatrixPower[{{2,2},{2,2}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Join[{1},(3*4^Range[25])/2] (* or *) Join[{1},NestList[4#&,6,25]] (* Harvey P. Dale, Feb 14 2012 *)

a(n)=3*4^n\2 \\ Charles R Greathouse IV, Oct 12 2015

print([int(6*4**(n-1)) for n in range(50)]) # Karl V. Keller, Jr., Aug 30 2021

a(n) = 4*a(n-1) for n > 1; a(0) = 1, a(1) = 6.
G.f.: (1+2*x)/(1-4*x).
a(n) = floor(6*4^(n-1)). - Karl V. Keller, Jr., Aug 30 2021
E.g.f.: (3*exp(4*x) - 1)/2. - Elmo R. Oliveira, Mar 31 2025

A247666 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 7-celled neighborhood.

Original entry on oeis.org

1, 7, 7, 25, 7, 49, 25, 103, 7, 49, 49, 175, 25, 175, 103, 409, 7, 49, 49, 175, 49, 343, 175, 721, 25, 175, 175, 625, 103, 721, 409, 1639, 7, 49, 49, 175, 49, 343, 175, 721, 49, 343, 343, 1225, 175, 1225, 721, 2863, 25, 175, 175, 625

Offset: 0

N. J. A. Sloane, Sep 22 2014

nonn

The neighborhood of a cell consists of the cell itself together with its six surrounding cells. A cell is ON at generation n iff an odd number of its neighbors were ON at the previous generation. We start with one ON cell.
This is the Run Length Transform of the sequence 1,7,25,103,409,1639,26215,... (almost certainly A102900).
This appears to be the same as the number of ON cells in a certain 2-D CA on the square grid in which the neighborhood of a cell is defined by f = 1/(x*y)+1/x+1/x*y+1/y+x/y+x+x*y, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation. Here is the neighborhood:
[X, 0, X]
[X, 0, X]
[X, X, X]
which contains a(1) = 7 ON cells.
This is the odd-rule cellular automaton defined by OddRule 557 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
Furthermore, this is also the number of ON cells in the 2-D CA on the square grid in which the neighborhood of a cell is defined by f = 1/(x*y)+1/x+1/y+1+y+x+x*y, with the same rule. Here is the neighborhood:
[0, X, X]
[X, X, X]
[X, X, 0]
- N. J. A. Sloane, Feb 19 2015
This is the odd-rule cellular automaton defined by OddRule 376 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
The partial sums are in A253767 in which the structure looks like an irregular stepped pyramid, apparently with a like-hexagonal base. - Omar E. Pol, Jan 29 2015

			From _Omar E. Pol_, Jan 29 2015: (Start)
May be arranged into blocks of sizes A011782:
1;
7;
7, 25;
7, 49, 25, 103;
7, 49, 49, 175, 25, 175, 103, 409;
7, 49, 49, 175, 49, 343, 175, 721, 25, 175, 175, 625, 103, 721, 409, 1639;
7, 49, 49, 175, 49, 343, 175, 721, 49, 343, 343, 1225, 175, 1225, 721, 2863, 25, 175, 175, 625, ...
It appears that right border gives A102900 without repetitions, see Comments section. [This is just a restatement of the fact that this sequence is the run length transform of what is presumably A102900. - _N. J. A. Sloane_, Feb 06 2015]
(End)
From _Omar E. Pol_, Mar 19 2015: (Start)
Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:
1;
..
7;
..
7;
25;
.........
7,    49;
25;
103;
...................
7,    49,  49, 175;
25,  175;
103;
409;
......................................
7,    49,  49, 175, 49, 343, 175, 721;
25,  175, 175, 625;
103, 721;
409;
1639;
...
Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).
(End)

Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, Illustrations of generations 0 to 4
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Cf. A102900, A071053, A160239, A247640.

C := f->`if`(type(f,`+`),nops(f),1);
f := 1+1/x+x+1/y+y+1/(x*y)+x*y;
g := n->expand(f^n) mod 2;
[seq(C(g(n)),n=0..100)];

A247666[n_] := Total[CellularAutomaton[{170, {2, {{1, 1, 0}, {1, 1, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 2]; Array[A247666, 52, 0] (* JungHwan Min, Sep 01 2016 *)
A247666L[n_] := Total[#, 2] & /@ CellularAutomaton[{170, {2, {{1, 1, 0}, {1, 1, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, n]; A247666L[51] (* JungHwan Min, Sep 01 2016 *)

a(n) = number of terms in expansion of f^n mod 2, where f = 1+1/x+x+1/y+y+1/(x*y)+x*y (mod 2);

A308890 Follow along the squares in the square spiral (as in A274640); in each square write the smallest positive number that a knight placed at that square cannot see.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 3, 4, 4, 3, 1, 2, 4, 3, 2, 1, 4, 4, 4, 3, 1, 2, 4, 4, 2, 1, 1, 1, 4, 4, 1, 1, 1, 3, 2, 4, 4, 1, 1, 1, 3, 3, 4, 3, 1, 1, 1, 3, 2, 4, 4, 2, 3, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 4, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 1

Offset: 1

N. J. A. Sloane, Jul 01 2019

nonn

Similar to A274640, except that here we consider the mex of squares that are a knight's moves rather than queen's moves.
Since there are at most 4 earlier cells in the spiral at a knight's move from any square, a(n) <= 5.
This is obtained by adding 1 to the terms of A308884. "Mex" here means minimal positive excluded value.

Cf. A247640, A274641, A308885-A308895.

A275667 Number of ON cells after n generations in a 2-dimensional "Odd-Rule" cellular automaton on triangular tiling.

Original entry on oeis.org

1, 3, 7, 9, 7, 21, 25, 27, 7, 21, 49, 63, 25, 75, 103, 81, 7, 21, 49, 63, 49, 147, 175, 189, 25, 75, 175, 225, 103, 309, 409, 243, 7, 21, 49, 63, 49, 147, 175, 189, 49, 147, 343, 441, 175, 525, 721, 567, 25, 75, 175, 225, 175, 525, 625, 675, 103, 309, 721

Offset: 0

Kovba Alexey, Aug 04 2016

Each triangular tile has 3 neighbors. A cell is ON in a given generation if and only if there was an odd number of ON cells among the three nearest neighbors in the preceding generation.
At the initial moment there is a single ON cell.
Given pattern replicates after a number of generations which is a power of 2 when a(n) = 7.
Number of cells on each even step minus one is divisible by 6.
By analogy with the Ekhad, Sloane, Zeilberger link, one may suppose that using ternary expansion of n, recurrence relations for a(n) can be obtained and proved.
From Andrey Zabolotskiy, Aug 04 2016: (Start)
If the first conjecture from the Formula section is true then the fact that the right border of the triangle (see Example) gives A000244 follows directly from it.
If the second conjecture is true then the numbers just before the right border give A102900.
Since the 7 cells which are ON at the beginning of every row are farther and farther away from each other, the n-th term of a row (with offset 0) is a(n)*7 for not very large n.
See also comments to A247666.
(End)
This is ETA rule 170. See the Sadat-Benedek reference for proof of pattern replication. - Paul Cousin, Apr 22 2025

			From _Omar E. Pol_, Aug 04 2016: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
3;
7, 9;
7, 21, 25, 27;
7, 21, 49, 63, 25, 75, 103, 81;
7, 21, 49, 63, 49, 147, 175, 189, 25, 75, 175, 225, 103, 309, 409, 243;
...
It appears that the right border gives A000244.
(End)

Paul Cousin, Table of n, a(n) for n = 0..16384
Paul Cousin, Illustration for n = 0..128
Paul Cousin, Elementary Triangular Automaton Rule 170
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015.
Alexey Kovba, Illustration for n = 0..5
MohammadReza Saadat and Benedek Nagy, Copy Machines - Self-reproduction with 2 States on Archimedean Tilings, Journal of Cellular Automata, vol. 17, pp. 221-249, 2023.
Index to sequences in the OEIS related to cellular automata

Cf. A160239 (square tiling analog), A247640, A247666 (hexagonal tiling analogs).
Cf. A000244, A011782, A102900, A382971.
Pattern replicating ETA rules: A383369 (rule 90).

a(0) = 1. Conjecture: a(2*t+1) = 3*a(t).

Conjectures: a(8*t+6) = 3*a(4*t+2) + 4*a(2*t), a(8*t+2) = 3*a(4*t) + 4*a(2*t), a(4*t) = a(2*t). These conjectured formulas together give recurrent relations for a(n) for any n. Also, obviously a(2*n) = A247666(n). - Andrey Zabolotskiy, Aug 04 2016

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.

A253100 Number of odd terms in f^n, where f = 1/(xy)+1/x+1/xy+1/y+x+x*y.

Original entry on oeis.org

1, 6, 6, 24, 6, 36, 24, 96, 6, 36, 36, 144, 24, 144, 96, 372, 6, 36, 36, 144, 36, 216, 144, 576, 24, 144, 144, 576, 96, 576, 372, 1416, 6, 36, 36, 144, 36, 216, 144, 576, 36, 216, 216, 864, 144, 864, 576, 2232, 24, 144, 144, 576, 144, 864, 576, 2304, 96, 576, 576, 2304, 372, 2232, 1416, 5340

Offset: 0

N. J. A. Sloane and Doron Zeilberger, Feb 19 2015

nonn

This is the number of ON cells in a certain two-dimensional cellular automaton in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there were an odd number of ON cells in the neighborhood at the previous generation.

This is the odd-rule cellular automaton defined by OddRule 347 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

			Here is the neighborhood f:
[X, 0, X]
[X, 0, X]
[X, X, 0]
which contains a(1) = 6 ON cells.

Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Cf. A253101. Similar to but different from A247640.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=1/(x*y)+1/x+1/x*y+1/y+x+x*y;
OddCA(f, 130);

(* f = A253101 *) f[n_] :=  2*(2-Sqrt[3])^n + 2*(2+Sqrt[3])^n - 2^n // Round; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *)

This is the Run Length Transform of A253101.

cp's OEIS Frontend

A164908 a(n) = (3*4^n - 0^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A247666 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 7-celled neighborhood.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A308890 Follow along the squares in the square spiral (as in A274640); in each square write the smallest positive number that a knight placed at that square cannot see.

Views

Author

Keywords

Comments

Crossrefs

A275667 Number of ON cells after n generations in a 2-dimensional "Odd-Rule" cellular automaton on triangular tiling.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A253100 Number of odd terms in f^n, where f = 1/(x*y)+1/x+1/x*y+1/y+x+x*y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A253100 Number of odd terms in f^n, where f = 1/(xy)+1/x+1/xy+1/y+x+x*y.