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.

Previous Showing 31-39 of 39 results.

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

Views

Author

N. J. A. Sloane, Mar 13 2021

Keywords

Comments

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.

Examples

			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.

Crossrefs

Cf. A151723, A170899,

A169778 a(n) = ceiling(A170905/2).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 5, 4, 7, 8, 1, 2, 3, 5, 5, 5, 9, 13, 9, 4, 7, 12, 14, 10, 16, 16, 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 4, 7, 12, 14, 14, 22, 34, 30, 14, 16, 28, 35, 25, 35, 32, 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21
Offset: 0

Views

Author

N. J. A. Sloane, May 08 2010

Keywords

Comments

The hexagon-based CA of A151723 has as symmetry group the dihedral group of order 12. Consider a one-twelfth slice; a(n) is the number of cells that are turned from OFF to ON at generation n.

Examples

			Illustration of generations 1 through 9:
.1
..2
...3
..4.4
.....5
..7.6.6
...7...7
..8.8.8.8
.........9
...
From _Omar E. Pol_, Feb 12 2013: (Start)
When written as a triangle from 1, the right border gives A011782 and row lengths give A011782.
1,
1,
1,2,
1,2,3,4,
1,2,3,5,5,4,7,8,
1,2,3,5,5,5,9,13,9,4,7,12,14,10,16,16,
1,2,3,5,5,5,9,13,9,5,9,15,19,17,21,29,17,4,7,12,14,14,22,34,30,14,16,28,35,25,35,32;
1,2,3,5,5,5,9,13,9,5,9,15,19,17,21,... (End)

Crossrefs

Cf. A151723, A151724, A170905.

Formula

A170905(n) = 2a(n) except A170905(1)=1.

A335795 First differences of A335794.

Original entry on oeis.org

0, 1, 6, 6, 18, 6, 30, 18, 54, 6, 30, 30, 78, 18, 90, 54, 150, 6, 30, 30, 78, 30, 126, 78, 222, 18, 90, 90, 234, 54, 246, 150, 390, 6, 30, 30, 78, 30, 126, 78, 222, 30, 126, 126, 318, 78, 366, 222, 606, 18, 90, 90, 234, 90, 378, 234, 666, 54, 246, 246, 630, 150, 630, 390, 966, 6, 30, 30, 78, 30, 126, 78, 222, 30, 126, 126, 318
Offset: 0

Views

Author

Cody B Duncan, Jun 23 2020

Keywords

Comments

Position of 6 occurs every 2^(n-1) + 1 for n > 0.
Position of 18 appears to occur according to the tetranacci numbers (cf. A000288): 4, 7, 13, 25, 49, ...

Examples

			When written as a triangle:
0
1
6
6 18
6 30 18 54
6 30 30 78 18 90  54 150
6 30 30 78 30 126 78 222 18 90 90 234 54 246 150 390
...

References

  • 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).

Crossrefs

Cf. A000288, A151723, A335794.

A163862 a(n) = A170907(2^n).

Original entry on oeis.org

1, 4, 13, 47, 176, 678, 2658, 10521, 41860, 166995, 667100, 2666651, 10663099, 42645332
Offset: 0

Views

Author

N. J. A. Sloane, May 11 2010

Keywords

Comments

This sequence determines the "lim sup" of the growth of A151723.

Crossrefs

Cf. A151723, A170907.

Formula

It appears that a(n)/4^n converges to a constant c which is roughly 0.635.

Extensions

a(8) onwards from David Applegate, May 12 2010

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.

A256139 First differences of A256138.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 36, 36, 28, 52, 60, 4, 12, 20, 36, 36, 36, 68, 100, 68, 28, 52, 92, 108, 76, 124, 124, 4, 12, 20, 36, 36, 36, 68, 100, 68, 36, 68, 116, 148, 132, 164, 228, 132, 28, 52, 92, 108, 108, 172, 268, 236, 108, 124, 220, 276, 196, 276, 252
Offset: 0

Views

Author

Omar E. Pol, Mar 20 2015

Keywords

Comments

Number of cells turned ON at n-th stage in the structure of A256138.
First differs from A169708 at a(11).

Examples

			Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,36,36,28,52,60;
4,12,20,36,36,36,68,100,68,28,52,92,108,76,124,124;
4,12,20,36,36,36,68,100,68,36,68,116,148,132,164,228,132,28,52,92,108,108,172,268,236,108,124,220,276,196,276,252;
...
It appears that the right border gives A173033.

Links

Crossrefs

Cf. A028399, A011782, A151723, A151724, A160721, A169708, A170898, A170905, A173033, A256138.

Formula

a(n) = 2*A151724(n+1)/3, n >= 1.

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

Views

Author

N. J. A. Sloane, Mar 14 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.

Links

Crossrefs

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

Views

Author

N. J. A. Sloane, Mar 14 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.

Links

Crossrefs

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

Views

Author

N. J. A. Sloane, Mar 14 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.

Links

Crossrefs

Cf. A151723, A342273.
Previous Showing 31-39 of 39 results.

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

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