A334169 a(n) is the number of ON-cells in the n-th full level of ON-cells of a triangular wedge in the hexagonal grid of A151723 (after 2^k >= n generations have been computed).

1, 2, 4, 6, 8, 10, 14, 16, 18, 26, 30, 32, 34, 50, 58, 62, 64, 66, 98, 114, 122, 126, 128, 130, 194, 226, 242, 250, 254, 256, 258, 386, 450, 482, 498, 506, 510, 512, 514, 770, 898, 962, 994, 1010, 1018, 1022, 1024, 1026, 1538, 1794, 1922, 1986, 2018, 2034, 2042, 2046, 2048, 2050, 3074, 3586, 3842

Offset

0,2

Comments

a(n) also is the distance of the full level of ON-cells from the apex of the triangular wedge. Note that 7 is the last generation modifying level 6 and, more generally for example, generation 2^m + 2^(m-1) + 1 is the last generation modifying level 2^m + 2, for m >= 1:

Level Generation ON-cells

1 1 1

2 2 1 1

3 3 1 0 1

4 4 1 1 1 1

5 5 1 0 0 0 1

6 7 1 1 1 1 1 1

7 7 1 0 1 0 1 0 1

8 8 1 1 1 1 1 1 1 1

9 9 1 0 0 0 0 0 0 0 1

10 13 1 1 1 1 1 1 1 1 1 1

...

Links

Table of n, a(n) for n=0..60.

Hartmut F. W. Hoft, Proof of positions of full levels

Formula

a(0) = 1; a(1) = 2, a(n) = 2 + Sum_{i=0..j} 2^(k-i-1), where k = floor((3 + sqrt(1 + 8*(n-2))/2) and j = n - 2 - (k-2)*(k-1)/2 for n >= 2.

Example

The sequence is the triangle below read by rows, where each row contains m-1 full levels of ON-cells from level 2^(m-1) + 2 through level 2^m, for m >= 2:
m\j   0    1    2    3    4    5    6    7    8
0:    1
1:    2
2:    4
3:    6    8
4:   10   14   16
5:   18   26   30   32
6:   34   50   58   62   64
7:   66   98  114  122  126  128
8:  130  194  226  242  250  254  256
9:  258  386  450  482  498  506  510  512
10: 514  770  898  962  994 1010 1018 1022 1024
...
A formula for the m-1 elements in positions (m, j), 0 <= j <= m-2, in each row m >= 2 is: b(m, j) = 2 + Sum_{k=0..j} 2^(m-k-1).

Mathematica

triangleRow[m_] := Map[2+Sum[2^(m-k-1), {k, 0, #}]&, Range[0, m-2]]/; m>=2
triangleRow[10] (* last line in triangle in Comments section *)
a334169[0]=1; a334169[1]=2; a334169[n_] := Module[{k, j}, k=Floor[(3 + Sqrt[1 + 8(n-2)])/2]; j = n - 2 - (k-2)(k-1)/2; 2 + Sum[2^(k-i-1), {i, 0, j}]]/; n>=2
Map[a334169, Range[0, 66]] (* sequence data *)

Crossrefs

Cf. A151723.

Sequence in context: A087370 A364158 A138929 * A180081 A340854 A367586

Adjacent sequences: A334166 A334167 A334168 * A334170 A334171 A334172

Keyword

nonn,tabf

Author

Hartmut F. W. Hoft, Apr 17 2020

Status

approved