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%I #21 Feb 01 2022 01:36:53
%S 1,3,7,11,15,19,23,27,31,35,43,47,51,55,59,63,67,83,91,95,99,107,111,
%T 115,119,123,127,131,163,179,187,191,195,211,219,223,227,235,239,243,
%U 247,251,255,259,323,355,371,379,383,387,419,435,443,447,451,467,475,479,483,491,495,499,503,507
%N 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).
%H Hartmut F. W. Hoft, <a href="/A334228/a334228.pdf">Plot of 3 rows of triangle L</a>
%F 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.
%F The formula for the count of ON-cells was verified through level 16384.
%e 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:
%e m/j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e 1: 1
%e 2: 3
%e 3: 7 11
%e 4: 15 19 23 27
%e 5: 31 35 43 47 51 55 59
%e 6: 63 67 83 91 95 99 107 111 115 119 123
%e 7: 127 131 163 179 187 191 195 211 219 223 227 235 239 243 247 251
%e ...
%e If T(i) = (i-2)*(i-1)/2 then row m >= 2 contains T(m) + 1 values.
%e 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:
%e m/j 1 2 3 4 5
%e 1: 4
%e 2: 4 4
%e 3: 4 8 4
%e 4: 4 16 8 4
%e 5: 4 32 16 8 4
%e ...
%e Applying function log_2(k) - 1 to an entry k in this triangle gives the corresponding entry in the triangle of A193592.
%e 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).
%t 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
%t a334228[n_] := Flatten[Map[row,Range[n]]] (* first n rows in triangle L *)
%t a334228[8] (* sequence data *)
%Y Cf. A002024, A002260, A004736, A151723, A193592.
%K nonn
%O 1,2
%A _Hartmut F. W. Hoft_, Apr 19 2020
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