A342278 -id:A342278 - cp's OEIS frontend

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

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

N. J. A. Sloane, Jan 10 2010

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

			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)

N. J. A. Sloane, Table of n, a(n) for n = 0..4092

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

N. J. A. Sloane, Mar 13 2021

nonn

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.

N. J. A. Sloane, Table of n, a(n) for n = 0..512

Cf. A151723, A151724, A170899, A342273, A342274.

This is A169787 with 1 subtracted from each term.

A342273 Consider the k-th row of triangle A170899 starting at the 3 in the middle of the row; the row from that point on converges to this sequence as k increases.

Original entry on oeis.org

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, 41, 21, 37, 51, 55, 71, 111, 127, 95, 79, 119, 179, 207, 219, 271, 279, 171, 81, 93, 137, 159, 195, 291, 363

Offset: 0

N. J. A. Sloane, Mar 13 2021

nonn

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.

Needs a bigger b-file.

			Row k=6 of A170899 breaks up naturally into 7 pieces:
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.
The last piece already matches the sequence for 16 terms. The number of matching terms doubles at each row.

N. J. A. Sloane, Table of n, a(n) for n = 0..255

Cf. A151723, A151724, A342272.

A342274 Consider the k-th row of triangle A170899, which has 2^k terms; discard the first quarter of the terms in the row; the remainder of the row converges to this sequence as k increases.

Original entry on oeis.org

4, 8, 14, 18, 18, 26, 42, 42, 26, 26, 46, 66, 70, 74, 98, 90, 42, 26, 46, 66, 74, 90, 138, 170, 134, 90, 114, 174, 194, 194, 226, 190, 74, 26, 46, 66, 74, 90, 138, 170, 138, 106, 146, 226, 274, 290, 346, 378, 262, 122, 114, 174, 210, 250, 362, 474

Offset: 0

N. J. A. Sloane, Mar 13 2021

nonn

This could be divided by 2 but then it would no longer be compatible with A342272 and A342273.

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.

			Row k=6 of A170899 breaks up naturally into 7 pieces:
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.
The penultimate piece matches the sequence for 8 terms. The number of matching terms doubles at each row.

Cf. A151723, A170899,

A342275 First differences of A170899.

Original entry on oeis.org

0, 1, -1, 1, 1, 1, -3, 1, 1, 2, 0, -1, 3, 1, -7, 1, 1, 2, 0, 0, 4, 4, -4, -5, 3, 5, 2, -4, 6, 0, -15, 1, 1, 2, 0, 0, 4, 4, -4, -4, 4, 6, 4, -2, 4, 8, -12, -13, 3, 5, 2, 0, 8, 12, -4, -16, 2, 12, 7, -10, 10, -3, -31, 1, 1, 2, 0, 0, 4, 4, -4, -4, 4

Offset: 0

N. J. A. Sloane, Mar 14 2021

sign

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, Table of n, a(n) for n = 0..4091

Cf. A151723, A170899.

A342276 First differences of A342272.

Original entry on oeis.org

1, 1, 2, 0, 0, 4, 4, -4, -4, 4, 6, 4, -2, 4, 8, -12, -12, 4, 6, 4, 0, 8, 16, 0, -18, -4, 16, 14, -4, 4, 12, -30, -28, 4, 6, 4, 0, 8, 16, 0, -16, 0, 20, 20, 4, 4, 24, -8, -50, -20, 16, 14, 4, 16, 40, 16, -40, -28, 28, 38, -6, 0, 14, -68, -60, 4, 6, 4

Offset: 0

N. J. A. Sloane, Mar 14 2021

sign

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, Table of n, a(n) for n = 0..511

Cf. A151723, A342272.

A342277 First differences of A342273.

Original entry on oeis.org

3, 5, 2, 0, 8, 12, -4, -12, 4, 16, 14, 0, 6, 20, -16, -36, -4, 16, 14, 4, 16, 40, 16, -36, -26, 28, 44, 6, 4, 28, -48, -86, -20, 16, 14, 4, 16, 40, 16, -32, -16, 40, 60, 28, 12, 52, 8, -108, -90, 12, 44, 22, 36, 96, 72, -64, -96, 28, 104, 26, -6, 28, -122, -188, -52, 16, 14, 4, 16, 40, 16, -32

Offset: 0

N. J. A. Sloane, Mar 14 2021

sign

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, Table of n, a(n) for n = 0..254

Cf. A151723, A342273.

cp's OEIS Frontend

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

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

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A342273 Consider the k-th row of triangle A170899 starting at the 3 in the middle of the row; the row from that point on converges to this sequence as k increases.

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A342274 Consider the k-th row of triangle A170899, which has 2^k terms; discard the first quarter of the terms in the row; the remainder of the row converges to this sequence as k increases.

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A342275 First differences of A170899.

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A342276 First differences of A342272.

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A342277 First differences of A342273.

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