cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jan 10 2010

Keywords

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

Examples

			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)

Links

Crossrefs

Cf. A139250, A151723, A151724, A170898.
A342272, A342273, A342274 are limiting sequences to which various parts of the rows of this triangle converge.

A342272 The rows of the triangle A170899 converge to this sequence.

Original entry on oeis.org

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
Offset: 0

Views

Author

N. J. A. Sloane, Mar 13 2021

Keywords

Comments

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.
A342273 is the limiting sequence for the part of the row of A170899 that start at the first "3".
Needs a bigger b-file.

Links

Crossrefs

Cf. A151723, A151724, A170899, A342273, A342274.
This is A169787 with 1 subtracted from each term.

A169786 Triangle read by rows: T(n,k) is number of cells that turn from OFF to ON at stage k of the growth of the obtuse triangle of hexagons described in the comment.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 5, 3, 3, 1, 1, 2, 3, 5, 5, 5, 9, 13, 7, 3, 5, 8, 6, 4, 1, 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 15, 3, 5, 8, 10, 10, 14, 22, 18, 8, 8, 13, 10, 5, 1, 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 5, 9, 15, 19, 19, 27, 43, 43, 25, 21, 37, 51, 47, 51, 63
Offset: 1

Views

Author

David Applegate and N. J. A. Sloane, May 12 2010

Keywords

Comments

Consider the tiling of the plane by hexagons, where each cell has 6 neighbors, as in the A151723, A151724, A170905, A170906.
Assume the hexagons are oriented so that each one has a pair of vertical edges.
We label the cells in the usual way by Eisenstein integers, complex numbers r+sw, where r,s in Z, w = exp(2 pi i / 3) (see Conway and Sloane, pp. 52-53).
Initially all cells are OFF.
For x >= 1, define a roughly triangular region B_x by declaring the cells {sw: s >= 1}, {r-w: r >= -1}, {x-1-i+iw: 0 <= i <= x-2}, {x-1-i+(i+1)w: 0 <= i <= x-3} to be permanently OFF.
In other words, B_x consists of 0 plus the cells {r+sw: 0 <= s <= x-3, 1 <= r <= x-s-2}.
At stage 1, the "corner" cell 0 is turned ON; thereafter, a cell in B_x is turned ON if it has exactly one ON neighbor. Once a cell is ON it stays ON.
T(n,k) is the number of cells in B_{2^n} that are turned from OFF to ON at stage k (1 <= k <= 2^n-1).
Row n has 2^n-1 terms.

Examples

			Example: B_8:
.W W W
..W 6 W W
...W 5 5 W W
....W 4 X 4 W W
.....W 3 3 4 X W W
......W 2 X 4 X 6 W W
.......1 2 3 4 5 6 7 W
........W W W W W W W
W = permanently OFF, X = OFF, ON cells are labeled with the stage at which they turned ON.
Triangle begins:
1,
1, 2, 1,
1, 2, 3, 5, 3, 3, 1,
1, 2, 3, 5, 5, 5, 9, 13, 7, 3, 5, 8, 6, 4, 1,
1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 15, 3, 5, 8, 10, 10, 14, 22, 18, 8, 8, 13, 10, 5, 1,
1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 5, 9, 15, 19, 19, 27, 43, 43, 25, 21, 37, 51, 47, 51, 63, 31, 3, 5, 8, 10, 10, 14, 22, 22, 14, 14, 24, 34, 36, 38, 50, 42, 16, 8, 13, 18, 20, 24, 36, 36, 20, 15, 21, 15, 6, 1,
1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 5, 9, 15, 19, 19, 27, 43, 43, 25, 21, 37, 51, 47, 51, 63, 33, 5, 9, 15, 19, 19, 27, 43, 43, 27, 27, 47, 67, 71, 75, 99, 91, 41, 21, 37, 51, 55, 71, 111, 127, 87, 59, 87, 125, 119, 119, 133, 63, 3, 5, 8, 10, 10, 14, 22, 22, 14, 14, 24, 34, 36, 38, 50, 46, 22, 14, 24, 34, 38, 46, 70, 86, 68, 46, 58, 88, 98, 98, 114, 92, 32, 8, 13, 18, 20, 24, 36, 44, 36, 28, 38, 58, 70, 74, 88, 88, 52, 23, 21, 31, 38, 44, 60, 64, 44, 30, 33, 21, 7, 1,
...
The rows converge to A169787.

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd ed., 1988, see pp. 52-53.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.