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
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
KEYWORD
nonn,tabf
AUTHOR
Hartmut F. W. Hoft, Apr 17 2020
STATUS
approved