A334228 a(n) is the n-th completed odd-numbered level 2k-1 containing k ON-cells in a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^j >= 2k-1 generations of the automaton in A151723 have been computed).

1, 3, 7, 11, 15, 19, 23, 27, 31, 35, 43, 47, 51, 55, 59, 63, 67, 83, 91, 95, 99, 107, 111, 115, 119, 123, 127, 131, 163, 179, 187, 191, 195, 211, 219, 223, 227, 235, 239, 243, 247, 251, 255, 259, 323, 355, 371, 379, 383, 387, 419, 435, 443, 447, 451, 467, 475, 479, 483, 491, 495, 499, 503, 507

Offset: 1

Author

Hartmut F. W. Hoft, Apr 19 2020

Keywords

• nonn

Examples

			The sequence appears to be the triangle L(m, j) below read by rows where each row m >= 2 contains the level numbers in the sequence between 2^m - 1 and 2^(m+1) - 5:
m/j  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
1:   1
2:   3
3:   7  11
4:  15  19  23  27
5:  31  35  43  47  51  55  59
6:  63  67  83  91  95  99 107 111 115 119 123
7: 127 131 163 179 187 191 195 211 219 223 227 235 239 243 247 251
...
If T(i) = (i-2)*(i-1)/2 then row m >= 2 contains T(m) + 1 values.
The difference structure of this triangle L(m, j) is given by the triangle P(m, 1) = 4 and P(m, j) = 2^(m+2-j), for 2 <= j <= m, of powers of 2 as follows:
m/j  1  2  3  4  5
1:   4
2:   4  4
3:   4  8  4
4:   4 16  8  4
5:   4 32 16  8  4
...
Applying function log_2(k) - 1 to an entry k in this triangle gives the corresponding entry in the triangle of A193592.
Going backwards in triangle P(m, j) from the row labeled m - 2, left to right up to its vertex, starting with 2^m - 1 and computing the cumulative differences using the entries in triangle P(m, j) produces the numbers in row m of triangle L(m, j).
		

Links

• Hartmut F. W. Hoft, Plot of 3 rows of triangle L

Crossrefs

Cf. A002024, A002260, A004736, A151723, A193592.

Programs

Mathematica

row[1] = 1; row[2] = 3; row[n_] := (2^n - 1) + Prepend[Accumulate[Flatten[Table[If[i==0||==j, 4, 2^(2+j-i)], {j, n-3, 0, -1}, {i, 0, j}]]], 0]/;n>=3
a334228[n_] := Flatten[Map[row,Range[n]]] (* first n rows in triangle L *)
a334228[8] (* sequence data *)

Formula

L(m, 0) = 2^(m-1); L(m, j) = 2^(m-1) + Sum_{i=k+1-j..k} P( t_1(i), t_2(i) ), for m >= 3 and for j=1..k, where k = (m-2)*(m-1)/2. Functions t_1(n) = floor(1/2 + sqrt(2*n)), A002024, and t_2(n) = binomial(floor(3/2 + sqrt(2*n)), 2) - n + 1, A004736, by Michael Somos, Jul 12 2003, are listed in triangle #7 in his link in A002260.

The formula for the count of ON-cells was verified through level 16384.