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
N. J. A. Sloane, Table of n, a(n) for n = 0..4092
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
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Jan 10 2010
STATUS
approved