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A170899 Triangle read by rows, obtained by subtracting 1 from the terms of A170898 and dividing by 2.

%I A170899 #34 Mar 14 2021 15:34:51
%S A170899 0,0,1,0,1,2,3,0,1,2,4,4,3,6,7,0,1,2,4,4,4,8,12,8,3,6,11,13,9,15,15,0,
%T A170899 1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,3,6,11,13,13,21,33,29,13,15,
%U A170899 27,34,24,34,31,0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,4,8,14
%N A170899 Triangle read by rows, obtained by subtracting 1 from the terms of A170898 and dividing by 2.
%C A170899 This sequence is essentially the number of cells that are turned ON at the n-th generation of a 30-degree sector of the hexagonal Ulam-Warburton cellular automaton in A151723. The cells on the six main diagonals are ignored, and the resulting counts have been divided by 12. - _N. J. A. Sloane_, Mar 13 2021
%C A170899 Row k has 2^k terms.
%C A170899 It would be nice to have a formula or recurrence for any of A170899, A342272-A342278, or any nontrivial relation between them. This might help to understand the fractal structure of the mysterious hexagonal Ulam-Warburton cellular automaton A151723. - _N. J. A. Sloane_, Mar 14 2021
%C A170899 It appears that this may also be regarded as a tetrahedron E(m,i,j), m>=0, i>=0, j>=0, in which the slice m is a triangle read by rows: R(i,j) in which row i has length A011782(i). - _Omar E. Pol_, Feb 13 2013
%C A170899 It appears that in the slice m (of the tetrahedron mentioned above) the differences between the first 2^(m-3) elements of row m-1 and the first 2^(m-3) elements of row m give the first 2^(m-3) elements of A169787, if m >= 3. Also it appears that the right border of slice m gives the first m powers of 2 together with 0. See the second arrangement in Example section. - _Omar E. Pol_, Mar 16 2013
%H A170899 N. J. A. Sloane, <a href="/A170899/b170899.txt">Table of n, a(n) for n = 0..4092</a>
%e A170899 Triangle begins:
%e A170899 0;
%e A170899 0,1;
%e A170899 0,1,2,3;
%e A170899 0,1,2,4,4,3,6,7;
%e A170899 0,1,2,4,4,4,8,12,8,3,6,11,13,9,15,15;
%e A170899 0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,3,6,11,13,13,21, 33,29,13,15,27,34,24,34,31;
%e A170899 0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32,3,6,11,13,13,21,33,29,17,21, 37,51,51,57,77,61,21,15,27,34,36,52,80,80,44,38,62,81,58, 73,63;
%e A170899 0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32,4,8,14,18,18,26,42,42,26,26, 46,66,70,74,98,90,40,20,36,50,54,70,110,126,86,58,86,124, 118,118,132,64,3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,25,21,37,51,55,71,111,127,91,65,93,137,143,147,175,127,37,15,27,34,36,52,80,80,56,56,88,126,136,150,192,172,84,46,62,81,90,124,184,196,124,96,139,183,131,152,127;
%e A170899 ...
%e A170899 From _Omar E. Pol_, Feb 13 2013 (Start):
%e A170899 When written as a tetrahedron the slices 0-7 are:
%e A170899 0;
%e A170899 ..
%e A170899 1;
%e A170899 0;
%e A170899 ..
%e A170899 1;
%e A170899 2;
%e A170899 3,0;
%e A170899 ....
%e A170899 1;
%e A170899 2;
%e A170899 4,4;
%e A170899 3,6,7,0;
%e A170899 ........
%e A170899 1;
%e A170899 2;
%e A170899 4,4;
%e A170899 4,8,12,8;
%e A170899 3,6,11,13,9,15,15,0;
%e A170899 ....................
%e A170899 1;
%e A170899 2;
%e A170899 4,4;
%e A170899 4,8,12,8;
%e A170899 4,8,14,18,16,20,28,16;
%e A170899 3,6,11,13,13,21,33,29,13,15,27,34,24,34,31,0;
%e A170899 .............................................
%e A170899 1;
%e A170899 2;
%e A170899 4,4;
%e A170899 4,8,12,8;
%e A170899 4,8,14,18,16,20,28,16;
%e A170899 4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32;
%e A170899 3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,21,15,27,34,36,52,80,80,44,38,62,81,58,73,63,0;
%e A170899 ..........................................................
%e A170899 1;
%e A170899 2;
%e A170899 4,4;
%e A170899 4,8,12,8;
%e A170899 4,8,14,18,16,20,28,16;
%e A170899 4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32;
%e A170899 4,8,14,18,18,26,42,42,26,26,46,66,70,74,98,90,40,20,36,50,54,70,110,126,86,58,86,124,118,118,132,64;
%e A170899 3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,25,21,37,51,55,71,111,127,91,65,93,137,143,147,175,127,37,15,27,34,36,52,80,80,56,56,88,126,136,150,192,172,84,46,62,81,90,124,184,196,124,96,139,183,131,152,127,0;
%e A170899 ..........................................................
%e A170899 (End)
%Y A170899 Cf. A139250, A151723, A151724, A170898.
%Y A170899 A342272, A342273, A342274 are limiting sequences to which various parts of the rows of this triangle converge.
%K A170899 nonn,tabf
%O A170899 0,6
%A A170899 _N. J. A. Sloane_, Jan 10 2010