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A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).

%I A334164 #28 May 29 2020 15:44:21
%S A334164 1,2,2,4,2,6,4,8,2,10,6,10,4,14,8,16,2,18,10,16,8,20,12,22,6,26,14,22,
%T A334164 8,30,16,32,2,34,18,28,16,34,18,32,14,40,22,34,16,42,24,44,10,50,26,
%U A334164 40,20,48,28,50,14,58,30,46,16,62,32,64
%N A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).
%C A334164 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.
%C A334164 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:
%C A334164 Index of the level: L(k, i) number of ON-cells: C(k, i)
%C A334164 k\i  0   1   2   3   4   5   6    k/i  0   1   2   3   4   5   6
%C A334164 2:   5                             2:  2
%C A334164 3:   9  13                         3:  2   4
%C A334164 4:  17  25  29                     4:  2   6   8
%C A334164 5:  33  49  57  61                 5:  2  10  14  16
%C A334164 6:  65  97 113 121 125             6:  2  18  26  30  32
%C A334164 7: 129 193 225 241 249 253         7:  2  34  50  58  62  64
%C A334164 8: 257 385 449 481 497 505 509     8:  2  66  98 114 122 126 128
%C A334164 ...
%C A334164 The pairs ( L(k, i), C(k, i) ), for 0 <= i <= k-2, define a line of slope 1/2 for each k >= 3.
%C A334164 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.
%C A334164 For triangle C(k, i): column 1 is A052548(n), n >= 1; column 2 is A164094(n), n >= 1.
%C A334164 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, ... .
%C A334164 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.
%C A334164 Sequence A334169 of even-numbered completed levels in which all cells are ON-cells is a subsequence of this sequence.
%H A334164 Hartmut F. W. Hoft, <a href="/A334164/a334164.pdf">Diagram of ON-cell counts</a>
%t A334164 (* a169781[] and support functions are defined in A169781 and create the list nTriangle *)
%t A334164 a334164[n_] := Module[{k, levels={}}, a169781[n]; For[k=1, k<=n, k++, AppendTo[levels, Count[nTriangle[[k]], 1] - 2]]; levels]/;(n>=3 && IntegerQ[Log[2,n]])
%t A334164 a334164[64] (* sequence data *)
%Y A334164 Cf. A000051, A036563, A052548, A083686, A151723, A164094, A181565, A195744, A196657, A206371, A334169.
%K A334164 nonn
%O A334164 1,2
%A A334164 _Hartmut F. W. Hoft_, Apr 17 2020