cp's OEIS Frontend

Number of Y-toothpicks added at n-th stage in the structure of A266532.

A simplified version of A160121.

a(n) is 1/6 of the difference between the number of Y-toothpicks added at n-th stage in the structure of A160120 and the number of Y-toothpicks added at n-th stage stage in the structure of A266532 (the outward version of A160120).

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).

To compute the binary expansion of a(n):

- we scan the binary digits of n from right to left,

- at some position k >= 0 (0 corresponding to the least significant bit):

- we count the number of 1's at positions >= k, say we have w 1's,

- if 2^w appears in the binary expansion of 2*n,

then we insert a 1,

otherwise we insert a 0,

- as we are considering an even automaton (with rule 2*n),

once scanning the leading 0's of n, we will only insert 0's,

- and the result will have finitely many 1's.

More formally: 2^k appears in the binary expansion of a(n) iff 2^A000120(floor(n/2^k)) appears in the binary expansion of 2*n.

Unlike A322050, this sequence contains only finitely many 1's. However, the Cellular Automaton and its counting sequences still admit a 2^n fractal structure (Cf. A322662). The subsequences L_n = {a(2^n), a(2^n+1), ... a(2^(n+1)-1)} appear to approach a limit sequence L_{oo}, starting with 4 ON cells. Of these 4, one is a "pioneer" at distance d*2^n from the origin, with d the distance of one knight step. The other three of four ON cells are due to retrogressive growth.

For any n >= 0, the n-th row:

- is palindromic,

- has A038573(n+1) terms,

- has leading term A006519(n+1),

- has central term A080079(n+1).

This sequence has similarities with A087734; here we reverse some consecutive bits, there we negate some consecutive bits.