A170899 -id:A170899 - cp's OEIS frontend

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

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.

A151724 First differences of A151723.

Original entry on oeis.org

0, 1, 6, 6, 18, 6, 18, 30, 42, 6, 18, 30, 54, 54, 42, 78, 90, 6, 18, 30, 54, 54, 54, 102, 150, 102, 42, 78, 138, 162, 114, 186, 186, 6, 18, 30, 54, 54, 54, 102, 150, 102, 54, 102, 174, 222, 198, 246, 342, 198, 42, 78, 138, 162, 162, 258, 402, 354, 162, 186

Offset: 0

David Applegate and N. J. A. Sloane, Jun 13 2009

nonn

			When written as a triangle:
0,
1,
6,
6,18,
6,18,30,42,
6,18,30,54,54,42,78,90,
6,18,30,54,54,54,102,150,102,42,78,138,162,114,186,186,
...
Right border gives 0 together with A068293. - _Omar E. Pol_, Mar 19 2015

S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 (see Example 6, page 224).

N. J. A. Sloane, Table of n, a(n) for n = 0..4095 [First 1024 terms from David Applegate and N. J. A. Sloane]
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
David Applegate and N. J. A. Sloane, Table of n, A151724(n), A151723(n) for n = 0..1025
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.

Cf. A151723, A170898 (after dividing by 6), A170899, A169759.

A170898 Triangle read by rows, obtained by dividing A151724 by 6.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 9, 9, 7, 13, 15, 1, 3, 5, 9, 9, 9, 17, 25, 17, 7, 13, 23, 27, 19, 31, 31, 1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41, 57, 33, 7, 13, 23, 27, 27, 43, 67, 59, 27, 31, 55, 69, 49, 69, 63, 1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41

Offset: 0

N. J. A. Sloane, Jan 10 2010

Row k has 2^k terms.

Right border gives the positive terms of A000225. - Omar E. Pol, Sep 28 2013

			Triangle begins:
1;
1,3;
1,3,5,7;
1,3,5,9,9,7,13,15;
1,3,5,9,9,9,17,25,17,7,13,23,27,19,31,31;
1,3,5,9,9,9,17,25,17,9,17,29,37,33,41,57,33,7,13,23,27,27,43,67,59,27,31,55,69,49,69,63;
...

N. J. A. Sloane, Table of n, a(n) for n = 0..1023

Cf. A169779 (partial sums).

Cf. A139250, A139251, A151723, A151724, A170899.

Equals A170905(n) - 1.

A342278 First differences of A342274.

Original entry on oeis.org

4, 6, 4, 0, 8, 16, 0, -16, 0, 20, 20, 4, 4, 24, -8, -48, -16, 20, 20, 8, 16, 48, 32, -36, -44, 24, 60, 20, 0, 32, -36, -116, -48, 20, 20, 8, 16, 48, 32, -32, -32, 40, 80, 48, 16, 56, 32, -116, -140, -8, 60, 36, 40, 112, 112, -48, -136, 0, 132, 64, -12, 28, -108, -256, -112, 20, 20, 8, 16, 48, 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, A342274.

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

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

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

A342275 First differences of A170899.

Views

Author

Keywords

Comments

Links

Crossrefs

A151724 First differences of A151723.

Views

Author

Keywords

Examples

References

Links

Crossrefs

A170898 Triangle read by rows, obtained by dividing A151724 by 6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A342278 First differences of A342274.

Views

Author

Keywords

Comments

Links

Crossrefs

A342276 First differences of A342272.

Views

Author

Keywords

Comments

Links

Crossrefs

A342277 First differences of A342273.

Views

Author

Keywords

Comments

Links

Crossrefs