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Partial sums of A170905.

First differs from both A169707 and A246335 at a(12).

First differs from the average of A169707 and A246335 at a(13).

Note that the above mentioned cellular automata work on the square grid.

A256139 gives the number of cells turned ON at the n-th stage.

Number of cells turned ON at n-th stage in one of the outside corners of an infinite hexagon-shaped structure on hexagonal grid.

For an animation see "The movie version" in Links section.

Total number of ON cells after n generations in one of the outside corners of an infinite hexagon-shaped structure on hexagonal grid.

For an animation see "The movie version" in Links section.

Partial sums of A256537.

See also the Formula section in A256537.

Compare A256138.

Also the number of ON cells after n generations in a knight's-move, one-neighbor, accumulative cellular automaton on the hexagonal lattice A_2. Define v(m)=2*sqrt(3)*[cos(m*Pi/3+Pi/6), sin(m*Pi/3+Pi/6)], vL(m)=2*v(m)+v(m+1), vR(m)=2*v(m)+v(m-1). The set of "knight's moves", M={vL(m):m=1,2,..6} U {vR(m):m=1,2,..6}, follows from an analogy between Z^2 and A_2. At each generation all ON cells remain ON while an OFF cell turns ON if and only if it has exactly one M-neighbor in the previous generation.

Fractal Structure Theorem (FST). A pair of lattice vectors M={v1,v2} generate a wedge, W = {x*v1 + y*v2 : x>=0, y>=0}. Define W-Subsets T_k such that T_{k+1}= T_k U { 2^n*v1 + v : v in T_k } U {2^n*v2 + v : v in T_k}, T_0 = { [0,0] }. The limit set T_{oo} is a fractal, and acquires the topology of a binary tree when points are connected by either v1 or v2. As a tree, T_k has height 2^k-1, with 2^k vertices at maximum depth, along a line in the direction v1-v2. Assume a one-M-neighbor, accumulative cellular automaton on W, where all vertices in T_k are ON. In the next generation, the front F_k={2^k*v1+m*(v2-v1) : 0<=m<=2^k} contains only two ON cells, {2^k*v1,2^k*v2}. The spacing, 2^k-1, is wide enough to turn ON two copies of T_k, one starting from each of the two ON cells in F_k. Thus T_{k+1} is also ON. Whenever only T_0 is ON as an initial condition, by induction, T_{oo} is ultimately ON.

The FST applies here to 12 distinct wedges: with {v1,v2}={vL(m), vR(m)} or with (v1,v2)={vL(m), vR(m+1)}, and m=1,2,..6. The triangle inequality ensures that paths including other vectors cannot reach the front F_k by generation 2^k. However, other vectors do generate retrogressive growth, which turns ON many additional cells.

The FST applies to a wide range of Cellular Automata. Wolfram's one-dimensional rule 90 gives the most elementary example where T_{oo} determines every ON cell. The tree structure T_{oo} also occurs with two-dimensional, accumulative, one-neighbor C.A. such as A151723, A319018, A147562. Also try: M={[0,1],[0,-1],[2,1],[-2,-1]}.

According to S. Ulam (cf. Links), some version of the FST was already known to J. Holladay circa 1960.

The FST implies scale resonance between this cellular automaton and the arrowed half hexagon tiling (cf. Links).

a(n) also is the distance of the full level of ON-cells from the apex of the triangular wedge. Note that 7 is the last generation modifying level 6 and, more generally for example, generation 2^m + 2^(m-1) + 1 is the last generation modifying level 2^m + 2, for m >= 1:

Level Generation ON-cells

1 1 1

2 2 1 1

3 3 1 0 1

4 4 1 1 1 1

5 5 1 0 0 0 1

6 7 1 1 1 1 1 1

7 7 1 0 1 0 1 0 1

8 8 1 1 1 1 1 1 1 1

9 9 1 0 0 0 0 0 0 0 1

10 13 1 1 1 1 1 1 1 1 1 1

...

High water marks in A151723.

Conjecture 1: Except for a(2^n + 1) = 2, n >= 1, for odd-numbered completed levels a lower bound of the ratio of ON-cells to the length of the level is (2^n + 2)/(3*2^(n+1) + 1) with limit 1/6, determined by the subsequence of levels starting with: 13, 25, 49, 97, 193, 385, 769, 1537, 3073, ..., and the associated ON-cell counts: 4, 6, 10, 18, 34, 66, 130, 258, 514, ..., as listed in the second column of each of the two triangles below.

The ON-cell counts for the indices in each row define a line of slope 1/2. The formula for the indices of levels in row k >= 2 is L(k, i) = 1 + Sum_{j = 0, ..., i} 2^(k-j), 0 <= i <= k - 2, and the formula for the associated numbers of ON-cells is C(k, i) = 2 + Sum_{j = 1..i} 2^(k-1-j), 0 <= i <= k - 2:

Index of the level: L(k, i) number of ON-cells: C(k, i)

k\i 0 1 2 3 4 5 6 k/i 0 1 2 3 4 5 6

2: 5 2: 2

3: 9 13 3: 2 4

4: 17 25 29 4: 2 6 8

5: 33 49 57 61 5: 2 10 14 16

6: 65 97 113 121 125 6: 2 18 26 30 32

7: 129 193 225 241 249 253 7: 2 34 50 58 62 64

8: 257 385 449 481 497 505 509 8: 2 66 98 114 122 126 128

...

The pairs ( L(k, i), C(k, i) ), for 0 <= i <= k-2, define a line of slope 1/2 for each k >= 3.

For triangle L(k, i): column 0 is A000051(n), n >= 2; column 1 is A181565(n), n >= 3; column 2 is A083686(n), n >= 2; columns 3 is A195744(n), n >= 1; column 4 is A206371(n), n >= 2; column 5 is A196657(n), n >= 1; the bounding diagonal is A036563(n), n >= 3.

For triangle C(k, i): column 1 is A052548(n), n >= 1; column 2 is A164094(n), n >= 1.

Conjecture 2: In an even-numbered completed level 2*n the fraction of ON-cells is bounded below by (23 * 2^n - 24)/(2^(n+5) - 36) with limit 23/32, determined by the subsequence of levels starting with: 28, 92, 220, 476, 988, 2012, 4060, ... .

There are 16 numbers less than 1000 that do not occur as the number of ON-cells in a completed level through level 16384: 136, 164, 330, 334, 402, 444, 526, 570, 598, 604, 614, 714, 740, 822, 832, 878.

Sequence A334169 of even-numbered completed levels in which all cells are ON-cells is a subsequence of this sequence.