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Triangle read by rows, obtained by subtracting 1 from the terms of A170898 and dividing by 2.
10

%I #34 Mar 14 2021 15:34:51

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

%C 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 Row k has 2^k terms.

%C 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 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 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 N. J. A. Sloane, <a href="/A170899/b170899.txt">Table of n, a(n) for n = 0..4092</a>

%e Triangle begins:

%e 0;

%e 0,1;

%e 0,1,2,3;

%e 0,1,2,4,4,3,6,7;

%e 0,1,2,4,4,4,8,12,8,3,6,11,13,9,15,15;

%e 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 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 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 ...

%e From _Omar E. Pol_, Feb 13 2013 (Start):

%e When written as a tetrahedron the slices 0-7 are:

%e 0;

%e ..

%e 1;

%e 0;

%e ..

%e 1;

%e 2;

%e 3,0;

%e ....

%e 1;

%e 2;

%e 4,4;

%e 3,6,7,0;

%e ........

%e 1;

%e 2;

%e 4,4;

%e 4,8,12,8;

%e 3,6,11,13,9,15,15,0;

%e ....................

%e 1;

%e 2;

%e 4,4;

%e 4,8,12,8;

%e 4,8,14,18,16,20,28,16;

%e 3,6,11,13,13,21,33,29,13,15,27,34,24,34,31,0;

%e .............................................

%e 1;

%e 2;

%e 4,4;

%e 4,8,12,8;

%e 4,8,14,18,16,20,28,16;

%e 4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32;

%e 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 ..........................................................

%e 1;

%e 2;

%e 4,4;

%e 4,8,12,8;

%e 4,8,14,18,16,20,28,16;

%e 4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32;

%e 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 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 ..........................................................

%e (End)

%Y Cf. A139250, A151723, A151724, A170898.

%Y A342272, A342273, A342274 are limiting sequences to which various parts of the rows of this triangle converge.

%K nonn,tabf

%O 0,6

%A _N. J. A. Sloane_, Jan 10 2010