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A170899
Triangle read by rows, obtained by subtracting 1 from the terms of A170898 and dividing by 2.
10
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, 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, 0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 14, 18, 16, 20, 28, 16, 4, 8, 14
OFFSET
0,6
COMMENTS
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
Row k has 2^k terms.
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
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
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
LINKS
EXAMPLE
Triangle begins:
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;
0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,3,6,11,13,13,21, 33,29,13,15,27,34,24,34,31;
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;
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;
...
From Omar E. Pol, Feb 13 2013 (Start):
When written as a tetrahedron the slices 0-7 are:
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;
....................
1;
2;
4,4;
4,8,12,8;
4,8,14,18,16,20,28,16;
3,6,11,13,13,21,33,29,13,15,27,34,24,34,31,0;
.............................................
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;
3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,21,15,27,34,36,52,80,80,44,38,62,81,58,73,63,0;
..........................................................
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;
3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,25,21,37,51,55,71,111,127,91,65,93,137,143,147,175,127,37,15,27,34,36,52,80,80,56,56,88,126,136,150,192,172,84,46,62,81,90,124,184,196,124,96,139,183,131,152,127,0;
..........................................................
(End)
CROSSREFS
A342272, A342273, A342274 are limiting sequences to which various parts of the rows of this triangle converge.
Sequence in context: A359941 A360302 A351939 * A221321 A367959 A179392
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jan 10 2010
STATUS
approved