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A334228
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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).
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1
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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.
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EXAMPLE
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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).
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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