cp's OEIS Frontend

Bisection of A323651. - Omar E. Pol, Mar 04 2019

First differs from A162795 at a(14), but it appears that then they share infinitely many terms. It appears that this is very close to A162795 rather than both A147562 and A169707.

The graphs of both A162795 and this sequence are intertwined.

Note that there are four main versions of this cellular automaton, depending on whether the wedges with the same rule are opposite or perpendicular and also depending on whether each mentioned version is represented by the "one-step rook" illustration or by the "one-step bishop" illustration. The four versions are represented by this sequence.

a(43) = 1729 is also the Hardy-Ramanujan number.

This sequence (for Ulam-Warburton) is analogous to A170927 (for toothpicks). Further, the lower limit of A147562(n)/n^2 evidently approaches twice the constant given in A195853.

Note that all values in this sequence are odd and that a(n)=2*a(n-1)+1 or a(n)=2*a(n-1)-1. - Robert Price, Aug 14 2015

It appears that these are the terms of A147562, A162795, A169707, A255366, A256250, A256260, whose indices have binary weight 1 or 2.

Evidently twice the lower limit of A139250(n)/n^2 and thus twice A195853.

A147562 = the partial sums of A147582, derived from the binary weight of n, wt() = A000120. A224513 = the partial sums of A224512, derived from the Gray code weight of n (number of 1's in the representation of n), gt() = A005811.

2^n-th terms = A002450 =(1, 5, 21, 85, 341, ...); as in A147562.

At the date of this submission, it's unknown if the terms represent a simple CA rule for the numbers of ON cells.

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).

Apparently the Gullwing sequence A187220 is the connection between the Ulam-Warburton C. A. (A147562) and the toothpick sequence A139250.

Note that the more complex structure is A187220, followed by A139250 and then A147562.

For another version which contains only positive terms and is more synchronous see the triangle A187564.

Apparently the Gullwing sequence A187220 is the connection between the Ulam-Warburton C. A. (A147562) and the toothpick sequence A139250.

Note that the more complex structure is A187220, followed by A139250 and then A147562.

For another version with minimal differences but asynchronous see A187223.