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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).
2
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
...
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
KEYWORD
nonn,tabf
AUTHOR
Hartmut F. W. Hoft, Apr 17 2020
STATUS
approved