A173530 -id:A173530 - cp's OEIS frontend

A231348 Number of triangles after n-th stage in a cellular automaton based in isosceles triangles of two sizes (see Comments lines for precise definition).

0, 1, 3, 7, 11, 15, 23, 33, 41, 45, 53, 65, 81, 91, 111, 133, 149, 153, 161, 173, 189, 201, 225, 253, 285, 295, 315, 343, 383, 405, 449, 495, 527, 531, 539, 551, 567, 579, 603, 631, 663, 675, 699, 731, 779, 807, 863, 923, 987, 997, 1017, 1045, 1085, 1113, 1169, 1233, 1313, 1335, 1379, 1439, 1527, 1573, 1665, 1759, 1823

Offset: 0

Omar E. Pol, Dec 15 2013

nonn

On the semi-infinite square grid the structure of this C.A. contains "black" triangles and "gray" triangles (see the Links section). Both types of triangles have two sides of length 5^(1/2). Every black triangle has a base of length 2 hence its height is 2 and its area is 2. Every gray triangle has a base of length 2^(1/2) hence its height is 3/(2^(1/2)) and its area is 3/2. Both types of triangles are arranged in the same way as the triangles of Sierpinski gasket (see A047999 and A006046). The black triangles are arranged in vertical direction. On the other hand the gray triangles are arranged in diagonal direction in the holes of the structure formed by the black triangles. Note that the vertices of all triangles coincide with the grid points.
The sequence gives the total number of triangles (black and gray) in the structure after n-th stage. A231349 (the first differences) gives the number of triangles added at n-th stage.
For a more complex structure see A233780.

			We start at stage 0 with no triangles, so a(0) = 0.
At stage 1 we add a black triangle, so a(1) = 1.
At stage 2 we add two black triangles, so a(2) = 1+2 = 3.
At stage 3 we add two black triangles and two gray triangles from the vertices of the master triangle, so a(3) = 3+2+2 = 7.
At stage 4 we add four black triangles, so a(4) = 7+4 = 11.
At stage 5 we add two black triangles and two gray triangles from the vertices of the master triangle, so a(5) = 11+2+2 = 15.
At stage 6 we add four black triangles and four gray triangles, so a(6) = 15+4+4 = 23.
At stage 7 we add four black triangles and six gray triangles, so a(7) = 23+4+6 = 33.
At stage 8 we add eight black triangles, so a(8) = 33+8 = 41.
And so on.
Note that always we add both black triangles and gray triangles except if n is a power of 2. In this case at stage 2^k we add only 2^k black triangles, for k >= 0.

Omar E. Pol, Illustration of the structure after 16 stages
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A047999, A006046, A139250, A151566, A160406, A173530, A182634, A194444, A220524, A220478, A230981, A231349, A233780.

A173531 a(0)=0: For n>0, a(n) = A060632(n)*A060632(n-1).

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 16, 8, 4, 8, 16, 16, 16, 32, 64, 16, 4, 8, 16, 16, 16, 32, 64, 32, 16, 32, 64, 64, 64, 128, 256, 32, 4, 8, 16, 16, 16, 32, 64, 32, 16, 32, 64, 64, 64, 128, 256, 64, 16, 32, 64, 64, 64, 128, 256, 128, 64, 128, 256, 256

Offset: 0

Omar E. Pol, Oct 10 2010

nonn

First differences of A173530.

Number of triangles (Or V-toothpicks, or L-toothpicks, etc.) added in the three-dimensional structure of A173530 at the n-th stage.

			If written as a triangle, begins:
0;
1;
2;
4,4;
4,8,16,8;
4,8,16,16,16,32,64,16;
4,8,16,16,16,32,64,32,16,32,64,64,64,128,256,32;
4,8,16,16,16,32,64,32,16,32,64,64,64,128,256,64,16,32,64,64,64,128,256,128,...

Michael De Vlieger, Table of n, a(n) for n = 0..16383
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.]
Christina Talar Bekaroğlu, Analyzing Dynamics of Larger than Life: Impacts of Rule Parameters on the Evolution of a Bug's Geometry, Master's thesis, Calif. State Univ. Northridge (2023). See p. 92.
Michael De Vlieger, Scatterplot of Log_2 a(n) for n = 1..1024.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A001316, A047999, A060632, A139250, A139251, A161207, A172311, A173530, A173532.

Prepend[Times @@@ Partition[Array[2^DigitCount[Floor[#/2], 2, 1] &, 120, 0], 2, 1], 0] (* Michael De Vlieger, Jan 11 2024 *)

A173532 a(n) = A173531(n) - A139251(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 4, 0, 4, 32, 0, 0, 0, 4, 4, 0, 4, 32, 12, 0, 4, 28, 24, 4, 40, 176, 0, 0, 0, 4, 4, 0, 4, 32, 12, 0, 4, 28, 24, 4, 40, 176, 28, 0, 4, 28, 24, 4, 40, 172, 72, 4, 36, 144, 116, 48, 256, 832, 0, 0, 0, 4, 4, 0, 4, 32, 12, 0, 4

Offset: 0

Omar E. Pol, Oct 10 2010

nonn

			If written as a triangle, begins:
0;
0;
0,0;
0,0,0,4;
0,0,0,4,4,0,4,32;
0,0,0,4,4,0,4,32,12,0,4,28,24,4,40,176;
0,0,0,4,4,0,4,32,12,0,4,28,24,4,40,176,28,0,4,28,24,4,40,172,72,4,36,144,116,48,256,832;

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. A139250, A139251, A161207, A172311, A173530, A173531.

cp's OEIS Frontend

A231348 Number of triangles after n-th stage in a cellular automaton based in isosceles triangles of two sizes (see Comments lines for precise definition).

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A173531 a(0)=0: For n>0, a(n) = A060632(n)*A060632(n-1).

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A173532 a(n) = A173531(n) - A139251(n).

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