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).

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, 8, 30, 16, 32, 2, 34, 18, 28, 16, 34, 18, 32, 14, 40, 22, 34, 16, 42, 24, 44, 10, 50, 26, 40, 20, 48, 28, 50, 14, 58, 30, 46, 16, 62, 32, 64

Offset: 1

Author

Hartmut F. W. Hoft, Apr 17 2020

Keywords

• nonn

Comments

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.

Links

• Hartmut F. W. Hoft, Diagram of ON-cell counts

Crossrefs

Cf. A000051, A036563, A052548, A083686, A151723, A164094, A181565, A195744, A196657, A206371, A334169.

Programs

Mathematica

(* a169781[] and support functions are defined in A169781 and create the list nTriangle *)
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]])
a334164[64] (* sequence data *)