%I #26 Jan 19 2022 21:55:37

%S 1,13,25,109,121,193,325,493,529,661,829,1129,1189,1405,1657,2101,

%T 2149,2281,2533,3133,3337,3709,4309,4909,5065,5449,5917,6757,6877,

%U 7381,7873,8845,8893,9025,9277,9877,10165,10849,11737

%N a(n) is to A151723(n+1) as A319018(n+1) is to A147562(n+1), n >= 0.

%C 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.

%C 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.

%C 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.

%C 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]}.

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

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

%H Bradley Klee, <a href="/A322662/a322662.png">Log-Periodic Coloring over Arrowed Half Hexagon tiling</a>.

%H Bradley Klee, <a href="/A322662/a322662_1.png">Log-Periodic Coloring to Stage 64</a>.

%H Bradley Klee, <a href="/A322662/a322662_2.png">T_n Tree Structure, n=1,2,3,4</a>.

%H Bradley Klee, <a href="http://demonstrations.wolfram.com/LimitPeriodicTilings/">Limit-Periodic Tilings</a>, Wolfram Demonstrations Project (2015).

%H S. M. Ulam, <a href="/A002858/a002858.pdf">On some mathematical problems connected with patterns of growth of figures</a>, pp. 216 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 [Annotated scanned copy]

%t HexStar=2*Sqrt[3]*{Cos[#*Pi/3+Pi/6],Sin[#*Pi/3+Pi/6]}&/@Range[0,5];

%t MoveSet=Join[2*HexStar+RotateRight[HexStar],2*HexStar+RotateLeft[HexStar]];

%t Clear@Pts;Pts[0] = {{0, 0}};

%t Pts[n_]:=Pts[n]=With[{pts=Pts[n-1]},Union[pts,Cases[Tally[Flatten[pts/.{x_,y_}:> Evaluate[{x,y}+#&/@MoveSet],1]],{x_,1}:>x]]];Length[Pts[#]]&/@Range[0,32]

%Y Hexagonal: A151723. Square: A319018, A147562. Tree: A006046, A267700, A038573. A322663.

%K nonn

%O 0,2

%A _Bradley Klee_, Dec 22 2018