A151907 -id:A151907 - cp's OEIS frontend

A151895 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.

0, 1, 5, 9, 13, 25, 29, 41, 53, 65, 85, 97, 117, 145, 149, 161, 173, 185, 213, 233, 261, 297, 333, 385, 429, 481, 533, 545, 573, 601, 629, 673, 717, 761, 837, 905, 989, 1033, 1085, 1145, 1197, 1257, 1309, 1337, 1397, 1457, 1525, 1625, 1669

Offset: 0

David Applegate and N. J. A. Sloane, Jul 30 2009

nonn

The cells are the squares of the standard square grid.
Cells are either OFF or ON, once they are ON they stay ON, and we begin in generation 1 with 1 ON cell.
Each cell has 4 neighbors, those that it shares an edge with. Cells that are ON at generation n all try simultaneously to turn ON all their neighbors that are OFF. They can only do this at this point in time; afterwards they go to sleep (but stay ON).
A square Q is turned ON at generation n+1 if:
a) Q shares an edge with one and only one square P (say) that was turned ON at generation n (in which case the two squares which intersect Q only in a vertex not on that edge are called Q's "outer squares"), and
b) Q's outer squares were not considered (that is, satisfied a)) in any previous generation, and
c) Q's outer squares are not prospective squares of the (n+1)st generation satisfying a).
Originally constructed in an attempt to explain the Holladay-Ulam CA shown in Fig. 2 of the 1962 Ulam article. However, as explained on page 222 of that article, the actual rule for that CA (see A151906, A151907) is different from ours.
A170896 and A267190 are also closely related cellular automata.
A151895 and A267190 first differ at n=17, when A267190 turns (12,2) ON even though its outer square (11,1) was considered (not turned ON) in a previous generation. - David Applegate, Jan 30 2016

D. 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.

David Applegate, Table of n, a(n) for n = 0..250
David Applegate, The movie version
David Applegate, Illustration of first 10 generations
David Applegate, Illustration of first 20 generations
David Applegate, Illustration of first 32 generations
David Applegate, Illustration of first 64 generations
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.], which is also available at arXiv:1004.3036v2
R. G. Schrandt and S. M. Ulam, On recursively defined geometric objects and patterns of growth [Link supplied by Laurinda J. Alcorn, Jan 09 2010.]
N. J. A. Sloane, Illustration of initial terms (concentrating on a 90-degree sector)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 [Annotated scanned copy]

See A170896, A170897 for the original Schrandt-Ulam version.

Cf. A151896 (the first differences), A139250, A151905, A151906, A151907, A267190, A267191.

We do not know of a recurrence or generating function.

Entry (including definition) revised by David Applegate and N. J. A. Sloane, Jan 21 2016

A151906 a(0) = 0, a(1) = 1; for n>1, a(n) = 8*A151905(n) + 4.

Original entry on oeis.org

0, 1, 4, 4, 4, 12, 4, 4, 12, 12, 12, 36, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 12, 12, 36, 36, 36, 108, 36, 36, 108, 108, 108, 324, 4, 4, 12, 12, 12, 36, 12, 12, 36, 36, 36, 108, 12, 12, 36, 36, 36, 108, 36, 36, 108, 108, 108

Offset: 0

David Applegate and N. J. A. Sloane, Jul 31 2009, Aug 03 2009

Consider the Holladay-Ulam CA shown in Fig. 2 and Example 2 of the Ulam article. Then a(n) is the number of cells turned ON in generation n.

			From _Omar E. Pol_, Apr 02 2018: (Start)
Note that this sequence also can be written as an irregular triangle read by rows in which the row lengths are the terms of A011782 multiplied by 3, as shown below:
0,1, 4;
4,4,12;
4,4,12,12,12,36;
4,4,12,12,12,36,12,12,36,36,36,108;
4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,324;
4,4,12,12,12,36,12,12,36,36,36,108,12,12,36,36,36,108,36,36,108,108,108,... (End)

S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

David Applegate, The movie version
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
N. J. A. Sloane, Illustration of initial terms (annotated copy of figure on p. 222 of Ulam)

Cf. A151904, A151905, A151907, A139250, A151895, A151896.

f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end;
A151904 := proc(n) local k,j; k:=floor(n/6); j:=n-6*k; (3^(wt(k)+f(j))-1)/2; end;
A151905 := proc (n) local k,j;
if (n=0) then 0;
elif (n=1) then 1;
elif (n=2) then 0;
else k:=floor( log(n/3)/log(2) ); j:=n-3*2^k; A151904(j); fi;
end;
A151906 := proc(n);
if (n=0) then 0;
elif (n=1) then 1;
else 8*A151905(n) + 4;
fi;
end;

wt[n_] := DigitCount[n, 2, 1];
f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
A151902[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
A151904[n_] := (3^A151902[n] - 1)/2;
A151905[n_] := Module[{k, j}, Switch[n, 0, 0, 1, 1, 2, 0, _, k = Floor[Log2[n/3]]; j = n - 3*2^k; A151904[j]]];
a[n_] := Switch[n, 0, 0, 1, 1, _, 8 A151905[n] + 4];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 16 2023, after Maple code *)

The three trisections are essentially A147582, A147582 and 3*A147582 respectively. More precisely, For t >= 1, a(3t) = a(3t+1) = A147582(t+1) = 4*3^(wt(t)-1), a(3t+2) = 4*A147582(t+1) = 4*3^wt(t). See A151907 for explanation.

A296510 Toothpick sequence on triangular grid (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 7, 13, 19, 25, 31, 41, 57, 77, 93, 103, 109, 119, 135, 159, 187, 219, 247, 279, 319, 369, 409, 431, 439, 449, 465, 489, 517, 549, 581, 621, 677, 751, 827, 891, 933, 969, 1009, 1071, 1147, 1237, 1317, 1405, 1507, 1629, 1725, 1775, 1789, 1799, 1815, 1839, 1867, 1899, 1931, 1971, 2027, 2101, 2177, 2241

Offset: 0

Omar E. Pol, Dec 14 2017

We use toothpicks of length 2, the same as the toothpick cellular automaton of A139250, but here we are on triangular grid, hence we have three axes, not two.
The Toothpicks are alternately arranged on the three axes in a rotating cycle.
a(n) gives the number of toothpicks in the structure after n-th stage.
A296511 (the first differences) gives the number of toothpicks added at n-th stage.
The structure reveals that some cellular automata that have recurrent periods can be represented by irregular triangles of first differences whose row lengths are the terms of A011782 multiplied by k (instead of powers of 2), where k is the length of their "word". In this case the word should be "abc", therefore k = 3. In the case of the cellular automaton with normal toothpicks (A139250) the word should be "ab", therefore k = 2.
For more information about the "word" of a cellular automaton see A296612.
Note that due to the unusual orientation of the polygons that are located on the edges of the structure, the image of this cellular automaton resembles the photo of an object that is rotating.
Note that between other polygons the structure contains the same "petals" as the floret pentagonal tiling.
Apparently the graph could be similar to the graph of A151907.

			After 49 stages in every 60-degree wedge of the mentioned dodecagon we can see six kind of closed regions as shown below:
----------------------------------------------------------------------------------
Polygon                    Sides's length  Perimeter   Area  Quantity  Total area
----------------------------------------------------------------------------------
Triangle                   [1,1,1]             3         1     100        100
Rhombus (diamond)          [2,2,2,2]           8         8       5         40
Trapeze                    [1,2,3,2]           8         8      35        280
Irregular pentagon (petal) [1,1,1,2,2]         7         7      58        406
Irregular pentagon         [1,1,3,2,4]        11        15       1         15
Hexagon                    [1,1,1,1,1,1]       6         6      20        120
----------------------------------------------------------------------------------
Subtotal per wedge                                             219        961
.
Then we have:
Subtotal of the six wedges                                    1308       5766
Shared triangle            [1,1,1]             3         1       2          2
----------------------------------------------------------------------------------
Total of the structure after 49 stages                        1306       5764

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
The Poly Pages, Polyiamonds
Rémy Sigrist, Illustration of the construction at generation 3*256
Wikipedia, Floret pentagonal tiling
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Cf. A151907, A160160, A296511 (first differences), A296612.

Cf. A160120 (word "a"), A139250 (word "ab"), A299476 (word "abcb"), A299478 (word "abcbc").

A151904 a(n) = (3^(wt(k)+f(j))-1)/2 if n = 6k+j, 0 <= j < 6, where wt = A000120, f = A151899.

Original entry on oeis.org

0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 4, 4, 13, 13, 13, 40, 13, 13, 40, 40, 40, 121, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 4, 4, 13, 13, 13, 40, 13, 13, 40, 40, 40, 121, 4, 4, 13, 13, 13, 40, 13, 13, 40, 40, 40

Offset: 0

N. J. A. Sloane, Jul 31 2009

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, A151899, A151902, A151905-A151907.

f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end;
A151904 := proc(n) local k,j; k:=floor(n/6); j:=n-6*k; (3^(wt(k)+f(j))-1)/2; end;

wt[n_] := DigitCount[n, 2, 1];
f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
A151902[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
a[n_] := (3^A151902[n] - 1)/2;
Table[a[n], {n, 0, 82}] (* Jean-François Alcover, Feb 16 2023 *)

a(n)=(3^(hammingweight(n\6)+[0,0,1,1,1,2][n%6+1])-1)/2 \\ Charles R Greathouse IV, Sep 26 2015

a(n) = (3^A151902(n)-1)/2.

A151905 a(0) = a(2) = 0, a(1) = 1; for n >= 3, n = 32^k+j, 0 <= j < 32^k, a(n) = A151904(j).

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 4, 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 4, 4, 13, 13, 13, 40, 13

Offset: 0

N. J. A. Sloane, Jul 31 2009

Consider the Holladay-Ulam CA shown in Fig. 2 and Example 2 of the Ulam article. Then a(n) is the number of cells turned ON in generation n in a 45-degree sector that are not on the main stem.

			If written as a triangle:
0,
1, 0,
0, 0, 1,
0, 0, 1, 1, 1, 4,
0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13,
0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40
0, 0, 1, 1, 1, 4, 1, 1, 4, 4, 4, 13, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 1, 1, 4, 4, 4, 13, 4, 4, 13, 13, 13, 40, 4, 4, 13, 13, 13, 40, 13, 13, 40, 40, 40, 121,
...
then the rows converge to A151904.

S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

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
N. J. A. Sloane, Illustration of initial terms (annotated copy of figure on p. 222 of Ulam)

Cf. A151904, A151906, A151907, A139250, A151895, A151896.

f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end;
A151904 := proc(n) local k,j; k:=floor(n/6); j:=n-6*k; (3^(wt(k)+f(j))-1)/2; end;
A151905 := proc (n) local k,j;
if (n=0) then 0;
elif (n=1) then 1;
elif (n=2) then 0;
else k:=floor( log(n/3)/log(2) ); j:=n-3*2^k; A151904(j); fi;
end;

wt[n_] := DigitCount[n, 2, 1];
f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
A151902[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
A151904[n_] := (3^A151902[n] - 1)/2;
a[n_] := Module[{k, j}, Switch[n, 0, 0, 1, 1, 2, 0, _, k = Floor[Log2[n/3]]; j = n - 3*2^k; A151904[j]]];
Table[a[n], {n, 0, 90}] (* Jean-François Alcover, Feb 16 2023, after Maple code *)

A151902 a(n) = wt(k) + f(j) if n = 6k+j, 0 <= j < 6, where wt() = A000120(), f() = A151899().

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 2, 2, 3, 2, 2, 3, 3, 3, 4, 1, 1, 2, 2, 2, 3, 2, 2, 3, 3, 3, 4, 2, 2, 3, 3, 3, 4, 3, 3, 4, 4, 4, 5, 1, 1, 2, 2, 2, 3, 2, 2, 3, 3, 3, 4, 2, 2, 3, 3, 3, 4, 3, 3, 4, 4, 4, 5, 2, 2, 3, 3, 3, 4, 3, 3, 4, 4, 4, 5, 3, 3, 4, 4, 4, 5, 4, 4, 5, 5, 5, 6, 1, 1, 2, 2, 2, 3, 2, 2, 3

Offset: 0

N. J. A. Sloane, Jul 31 2009

nonn

Cf. A000120, A151899, A151904-A151907.

f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end;
A151902 := proc(n) local k,j; k:=floor(n/6); j:=n-6*k; wt(k)+f(j); end;

wt[n_] := DigitCount[n, 2, 1];
f[n_] := {0, 0, 1, 1, 1, 2}[[Mod[n, 6] + 1]];
a[n_] := wt[Floor[n/6]] + f[n - 6 Floor[n/6]];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Feb 16 2023 *)

A151899 Period 6: repeat [0, 0, 1, 1, 1, 2].

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1

Offset: 0

N. J. A. Sloane, Jul 31 2009

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A151902, A151904, A151905, A151906, A151907.

[Abs( ((1-n) mod 3) - ((1+n) mod 2) ) : n in [0..100]]; // Wesley Ivan Hurt, Aug 20 2014

f := proc(n) local j; j:=n mod 6; if (j<=1) then 0 elif (j<=4) then 1 else 2; fi; end;
A151899:=n->[0, 0, 1, 1, 1, 2][(n mod 6)+1]: seq(A151899(n), n=0..100); # Wesley Ivan Hurt, Jun 20 2016

Table[Abs[Mod[-n + 1, 3] - Mod[n + 1, 2]], {n, 0, 100}] (* Wesley Ivan Hurt, Aug 20 2014 *)
CoefficientList[Series[(x^2 + x^3 + x^4 + 2 x^5)/(1 - x^6), {x, 0, 100}], x] (* Wesley Ivan Hurt, Aug 20 2014 *)
LinearRecurrence[{0, 0, 0, 0, 0, 1},{0, 0, 1, 1, 1, 2},105] (* Ray Chandler, Aug 26 2015 *)

a(n)=[0,0,1,1,1,2][n%6+1]; \\ Joerg Arndt, Aug 25 2014

a(n) = 5/6 - cos(Pi*n/3)/3 - sin(Pi*n/3)/sqrt(3) - cos(2*Pi*n/3)/3 - sin(2*Pi*n/3)/sqrt(3) - (-1)^n/6. - R. J. Mathar, Oct 08 2011
G.f.: (x^2+x^3+x^4+2*x^5)/(1-x^6); a(n) = abs( mod(1-n,3) - mod(1+n,2) ). - Wesley Ivan Hurt, Aug 20 2014
a(n) = a(n-6) for n>5. - Wesley Ivan Hurt, Jun 20 2016

A373226 Number of points in a diagonal Vicsek fractal subset of an n X n square.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 10, 13, 18, 25, 26, 29, 30, 33, 38, 45, 46, 49, 50, 53, 58, 65, 74, 85, 90, 97, 110, 125, 126, 129, 130, 133, 138, 145, 146, 149, 150, 153, 158, 165, 174, 185, 190, 197, 210, 225, 226, 229, 230, 233

Offset: 0

Sidharth Ghoshal, May 28 2024

			For n=3, a(3)=5 by counting x's in the following ASCII pattern:
  x.x
  .x.
  x.x
For n=4, a(4)=6 by counting x's in the following ASCII pattern:
  x...
  .x.x
  ..x.
  .x.x
For n=5, a(5)=9 by counting x's in the following ASCII pattern:
  x...x
  .x.x.
  ..x..
  .x.x.
  x...x
Generally for odd n, one can construct a diagonal Vicsek fractal on a 3^k X 3^k matrix such that 3^k >= n: place an n X n square in the center and count the x's.
For even n, there are 4 ways to most "centrally" place an n X n square; however, due to 4-fold symmetry of the diagonal Vicsek fractal they result in the same value. In our ASCII convention above we use the top-left selection.

Wikipedia, Vicsek fractal.

Cf. A151907 (conjectured to be only the odd terms).

import copy
def combine(matrix): #accepts a matrix of matrices and fuses them
    aggregate = []
    for row in matrix:
        for j in range(0,len(matrix[0][0])):
            agg_row = []
            for block in row:
                agg_row += block[j]
            aggregate.append(agg_row)
    return aggregate
def descend(seed, zero, source, bound): #general fractal constructor
    for i in range(0, bound):
        for q in range(0, len(source)):
            for r in range(0, len(source[q])):
                if source[q][r] == 1:
                    source[q][r] = copy.deepcopy(seed)
                if source[q][r] == 0:
                    source[q][r] = copy.deepcopy(zero)
        source = combine(source) #fuse it up
    return source
def count(matrix, bound):
    counter = 0
    center_x, center_y = len(matrix)//2, len(matrix)//2
    shift_limit = bound//2
    if len(matrix)%2 == 1 and bound % 2 == 1:
        for i in range(-shift_limit, shift_limit+1):
            for j in range(-shift_limit, shift_limit+1):
                if matrix[center_x+i][center_y+j] == 1:
                    counter += 1
    if len(matrix)%2 == 1 and bound % 2 == 0:
        for i in range(-shift_limit, shift_limit):
            for j in range(-shift_limit, shift_limit):
                if matrix[center_x+i][center_y+j] == 1:
                    counter += 1
    return counter
seed = [[1,0,1],[0,1,0],[1,0,1]]
source = [[1]]
zero = [[0,0,0],[0,0,0],[0,0,0]]
#example with n=5
n=5
print(count(descend(seed,zero,source,2),5)) #this constructs a 3^2 X 3^2 matrix and counts the center 5 X 5 matrix

Conjecture: a(n) = O(n^log_3(5)). This is motivated by the log_3(5) Hausdorff dimension of the diagonal Vicsek fractal.
Conjecture: a(3n) = 5*a(n). This is motivated by the recursive construction of the diagonal Vicsek fractal.
Conjecture: a(2n+1) = A151907(n)

cp's OEIS Frontend

A151895 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.

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A151906 a(0) = 0, a(1) = 1; for n>1, a(n) = 8*A151905(n) + 4.

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A296510 Toothpick sequence on triangular grid (see Comments lines for definition).

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A151904 a(n) = (3^(wt(k)+f(j))-1)/2 if n = 6k+j, 0 <= j < 6, where wt = A000120, f = A151899.

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A151905 a(0) = a(2) = 0, a(1) = 1; for n >= 3, n = 3*2^k+j, 0 <= j < 3*2^k, a(n) = A151904(j).

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A151902 a(n) = wt(k) + f(j) if n = 6k+j, 0 <= j < 6, where wt() = A000120(), f() = A151899().

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Maple

Mathematica

A151899 Period 6: repeat [0, 0, 1, 1, 1, 2].

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Magma

Maple

Mathematica

PARI

Formula

A373226 Number of points in a diagonal Vicsek fractal subset of an n X n square.

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A151905 a(0) = a(2) = 0, a(1) = 1; for n >= 3, n = 32^k+j, 0 <= j < 32^k, a(n) = A151904(j).