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 1-9 of 9 results.

A169779 Partial sums of A170898.

Original entry on oeis.org

1, 2, 5, 6, 9, 14, 21, 22, 25, 30, 39, 48, 55, 68, 83, 84, 87, 92, 101, 110, 119, 136, 161, 178, 185, 198, 221, 248, 267, 298, 329, 330, 333, 338, 347, 356, 365, 382, 407, 424, 433, 450, 479, 516, 549, 590, 647, 680, 687, 700, 723, 750, 777, 820, 887, 946, 973, 1004, 1059
Offset: 0

Views

Author

N. J. A. Sloane, May 10 2010

Keywords

Comments

On the hexagonal grid consider an infinite 60-degree wedge. A cell is turned ON if exactly one of its six neighbors is ON. We start with a single ON cell. An ON cell remains ON forever. The sequence counts the total number of ON states after n generations. The structure is also the tree that arises from one of the six spokes of the structure of A151723. For n >> 1 the structure looks like a quadrilateral formed by two scalene right triangles which are joined at their hypotenuses. - Omar E. Pol, Mar 06 2013

Links

Crossrefs

Cf. A151723, A169780.

Formula

a(n) = (A151723(n+2) - 1)/6. - Omar E. Pol, Mar 06 2013

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.

A151723 Total number of ON states after n generations of cellular automaton based on hexagons.

Original entry on oeis.org

0, 1, 7, 13, 31, 37, 55, 85, 127, 133, 151, 181, 235, 289, 331, 409, 499, 505, 523, 553, 607, 661, 715, 817, 967, 1069, 1111, 1189, 1327, 1489, 1603, 1789, 1975, 1981, 1999, 2029, 2083, 2137, 2191, 2293, 2443, 2545, 2599, 2701, 2875, 3097, 3295
Offset: 0

Views

Author

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

Keywords

Comments

Analog of A151725, but here we are working on the triangular lattice (or the A_2 lattice) where each hexagonal cell has six neighbors.
A cell is turned ON if exactly one of its six neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
It would be nice to find a recurrence for this sequence!
Has a behavior similar to A182840 and possibly to A182632. - Omar E. Pol, Jan 15 2016

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).

Links

Crossrefs

Cf. A147562, A151724, A151725, A161206, A161644, A169779, A169780, A170898, A170905, A182632, A182840, A256536, A256537.

Programs

  • Mathematica
    A151723[0] = 0; A151723[n_] := Total[CellularAutomaton[{10926, {2, {{2, 2, 0}, {2, 1, 2}, {0, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{{n - 1}}}], 2]; Array[A151723, 47, 0](* JungHwan Min, Sep 01 2016 *)
A151723L[n_] := Prepend[Total[#, 2] & /@ CellularAutomaton[{10926, {2, {{2, 2, 0}, {2, 1, 2}, {0, 2, 2}}}, {1, 1}}, {{{1}}, 0}, n - 1], 0]; A151723L[46] (* JungHwan Min, Sep 01 2016 *)

Formula

a(n) = 6*A169780(n) - 6*n + 1 (this is simply the definition of A169780).
a(n) = 1 + 6*A169779(n-2), n >= 2. - Omar E. Pol, Mar 19 2015
It appears that a(n) = a(n-2) + 3*(A256537(n) - 1), n >= 3. - Omar E. Pol, Apr 04 2015

Extensions

Edited by N. J. A. Sloane, Jan 10 2010

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

Views

Author

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

Keywords

Examples

			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

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).

Links

Crossrefs

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

A170905 Consider the hexagonal cellular automaton defined in A151723, A151724; a(n) = number of cells that go from OFF to ON at stage n, if we only look at a 60-degree wedge (including the two bounding edges).

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 10, 10, 8, 14, 16, 2, 4, 6, 10, 10, 10, 18, 26, 18, 8, 14, 24, 28, 20, 32, 32, 2, 4, 6, 10, 10, 10, 18, 26, 18, 10, 18, 30, 38, 34, 42, 58, 34, 8, 14, 24, 28, 28, 44, 68, 60, 28, 32, 56, 70, 50, 70, 64, 2, 4, 6, 10, 10, 10, 18, 26, 18, 10, 18, 30, 38, 34, 42
Offset: 0

Views

Author

N. J. A. Sloane, Jan 22 2010

Keywords

Examples

			From _Omar E. Pol_, Feb 12 2013: (Start)
When written as a triangle starting from 1, the right border gives A000079 and row lengths give A011782.
1;
2;
2,4;
2,4,6,8;
2,4,6,10,10,8,14,16;
2,4,6,10,10,10,18,26,18,8,14,24,28,20,32,32;
2,4,6,10,10,10,18,26,18,10,18,30,38,34,42,58,34,8,14,24,28,28,44,68,60,28,32,56,70,50,70,64;
2,4,6,10,10,10,18,26,18,10,18,30,38,34,42,...
... (End)

Links

Crossrefs

Cf. A151723, A151724, A170898, A169778, A169780 (partial sums).

Formula

a(n) = A170898(n-2) + 1 for n >= 2.
a(n) = 2*A169778(n) for n != 1.

A256138 Total number of ON states after n generations of cellular automaton of A151723 based on hexagons, if we only look at two opposite 120-degree wedges, including the central cell.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 57, 85, 89, 101, 121, 157, 193, 221, 273, 333, 337, 349, 369, 405, 441, 477, 545, 645, 713, 741, 793, 885, 993, 1069, 1193, 1317, 1321, 1333, 1353, 1389, 1425, 1461, 1529, 1629, 1697, 1733, 1801, 1917, 2065, 2197, 2361, 2589, 2721, 2749, 2801, 2893, 3001, 3109, 3281, 3549, 3785, 3893, 4017, 4237, 4513, 4709, 4985, 5237
Offset: 1

Views

Author

Omar E. Pol, Mar 20 2015

Keywords

Comments

First differs from both A169707 and A246335 at a(12).
First differs from the average of A169707 and A246335 at a(13).
Note that the above mentioned cellular automata work on the square grid.
A256139 gives the number of cells turned ON at the n-th stage.

Links

Crossrefs

Cf. A151723, A169707, A169779, A169780, A170898, A246333, A246335, A256139.

Formula

a(n) = 1 + 2*(A151723(n) - 1)/3 = 1 - 4*n + 4*A169780(n).
a(n) = 1 + 4*A169779(n-2), n >= 2.
a(n) = A151723(n) - 2*A169779(n-2), n >= 2.

A256537 First differences of corner sequence A256536 associated with A151723.

Original entry on oeis.org

1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41, 57, 33, 9, 17, 29, 37, 37, 53, 85, 85, 49, 41, 73, 101, 93, 101, 125, 65, 9, 17, 29, 37, 37, 53, 85, 85, 53, 53, 93, 133, 141, 149, 197, 181, 81, 41, 73, 101, 109, 141, 221, 253, 173, 117, 173, 249, 237, 237, 265, 129
Offset: 1

Views

Author

Omar E. Pol, Apr 02 2015

Keywords

Comments

Number of cells turned ON at n-th stage in one of the outside corners of an infinite hexagon-shaped structure on hexagonal grid.
For an animation see "The movie version" in Links section.

Examples

			Written as an irregular triangle in which the row lengths are the absolute values of the terms of A141531, the sequence begins:
  1;
  3;
  5;
  9, 9;
  9, 17, 25, 17;
  9, 17, 29, 37, 33, 41, 57, 33;
  9, 17, 29, 37, 37, 53, 85, 85, 49, 41, 73, 101, 93, 101, 125, 65;
  9, 17, 29, 37, 37, 53, 85, 85, 53, 53, 93, 133, 141, 149, 197, 181, 81, 41, 73, 101, 109, 141, 221, 253, 173, 117, 173, 249, 237, 237, 265, 129;
  ...
It appears that the right border gives A083318, whose representation in base 2 gives A000533.

Links

Crossrefs

Cf. A000533, A083318, A141531, A151723, A151724, A169779, A170898, A256536.

Formula

a(1) = 1; a(2) = 3.
It appears that a(n) = 1 + (A151724(n) + A151724(n-1))/3, n >= 3.
It appears that a(n) = 1 + (A151723(n) - A151723(n-2))/3, n >= 3.
It appears that a(n) = 1 + 2*(A170898(n-2) + A170898(n-3)), n >= 3.
a(3) = 5.
It appears that a(n) = 1 + 2*(A169779(n-2) - A169779(n-4)), n >= 4.

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.

A322663 First differences of A322662 divided by 12.

Original entry on oeis.org

1, 1, 7, 1, 6, 11, 14, 3, 11, 14, 25, 5, 18, 21, 37, 4, 11, 21, 50, 17, 31, 50, 50, 13, 32, 39, 70, 10, 42, 41, 81, 4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, 18, 37, 64, 151, 41, 80, 126, 131, 29
Offset: 1

Views

Author

Bradley Klee, Dec 22 2018

Keywords

Comments

Unlike A322050, this sequence contains only finitely many 1's. However, the Cellular Automaton and its counting sequences still admit a 2^n fractal structure (Cf. A322662). The subsequences L_n = {a(2^n), a(2^n+1), ... a(2^(n+1)-1)} appear to approach a limit sequence L_{oo}, starting with 4 ON cells. Of these 4, one is a "pioneer" at distance d*2^n from the origin, with d the distance of one knight step. The other three of four ON cells are due to retrogressive growth.

Examples

			Written as a 2^k triangle:
1,
1, 7,
1, 6,  11, 14,
3, 11, 14, 25, 5,  18, 21, 37,
4, 11, 21, 50, 17, 31, 50, 50, 13, 32, 39, 70,  10, 42, 41, 81,
4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, ...

Crossrefs

Hexagonal: A151724, A170898, A256537. Square: A147582, A147610, A048883; A319019, A322050, A322049. Lower Bound: A038573.

Programs

  • Mathematica
    HexStar=2*Sqrt[3]*{Cos[#*Pi/3+Pi/6],Sin[#*Pi/3+Pi/6]}&/@Range[0,5];
MoveSet2 =Join[2*HexStar+RotateRight[HexStar],2*HexStar+RotateLeft[HexStar]];
Clear@Pts;Pts[0] = {{0, 0}};
Pts[n_]:=Pts[n]=With[{pts=Pts[n-1]},Union[pts,Cases[Tally[Flatten[pts/.{x_,y_}:> Evaluate[{x,y}+#&/@MoveSet2],1]],{x_,1}:>x]]];
Abs[(1/12)*Subtract@@#&/@Partition[Length[Pts[#]]&/@Range[0,32],2,1]]

Formula

a(n) = (A322662(n)-A322662(n-1))/12.
Showing 1-9 of 9 results.

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