A256260 -id:A256260 - cp's OEIS frontend

A256261 First differences of A256260.

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 28, 20, 44, 68, 60, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 92, 116, 140, 164, 124, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 92, 116, 140, 164, 124, 20, 44, 68, 92, 116, 140, 164, 188, 212, 236, 260, 284, 308, 332, 356, 252, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 92, 116, 140

Offset: 0

Omar E. Pol, Mar 30 2015

First 27 terms agree with A169708. Both sequences share infinitely many terms.

			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;
4,12,20,28,20,44,68,60;
4,12,20,28,20,44,68,60,20,44,68,92,116,140,164,124;
4,12,20,28,20,44,68,60,20,44,68,92,116,140,164,124,20,44,68,92,116,140,164,188,212,236,260,284,308,332,356,252;
...
It appears that the row sums give A000302.
It appears that the right border gives A173033.

Cf. A000302, A011782, A139251, A147582, A169708, A173033, A256258, A256251, A256260, A256263, A256264, A256265.

a(n) = 4*A256263(n), n >= 1.

A160172 T-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 4, 9, 18, 27, 36, 49, 74, 95, 104, 117, 142, 167, 192, 229, 302, 359, 368, 381, 406, 431, 456, 493, 566, 627, 652, 689, 762, 835, 908, 1017, 1234, 1399, 1408, 1421, 1446, 1471, 1496, 1533, 1606, 1667, 1692, 1729, 1802, 1875, 1948, 2057, 2274, 2443, 2468

Offset: 0

Omar E. Pol, Jun 01 2009

A T-toothpick is formed from three toothpicks of equal length, in the shape of a T. There are three endpoints. We call the middle of the top toothpick the pivot point.
We start at round 0 with no T-toothpicks.
At round 1 we place a T-toothpick anywhere in the plane.
At round 2 we place three other T-toothpicks.
And so on...
The rule for adding a new T-toothpick is the following. A new T-toothpick is added at any exposed endpoint, with the pivot point touching the endpoint and so that the crossbar of the new toothpick is perpendicular to the exposed end.
The sequence gives the number of T-toothpicks after n rounds. A160173 (the first differences) gives the number added at the n-th round.
See the entry A139250 for more information about the toothpick process and the toothpick propagation.
On the infinite square grid a T-toothpick can be represented as a square polyedge with three components from a central point: two consecutive components on the same straight-line and a centered orthogonal component.
If the T-toothpick has three components then at the n-th round the structure is a polyedge with 3*a(n) components.
From Omar E. Pol, Mar 26 2011: (Start)
For formula and more information see the Applegate-Pol-Sloane paper, chapter 11, "T-shaped toothpicks". See also A160173.
Also, this sequence can be illustrated using another structure in which every T-toothpick is replaced by an isosceles right triangle. (End)
The structure is very distinct but the graph is similar to the graphs from the following sequences: A147562, A160164, A162795, A169707, A187220, A255366, A256260, at least for the known terms from Data section. - Omar E. Pol, Nov 24 2015
Shares with A255366 some terms with the same index, for example the element a(43) = 1729, the Hardy-Ramanujan number. - Omar E. Pol, Nov 25 2015

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.],
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A147562, A160120, A160160, A160164, A160170, A160173, A160406, A160408, A160426, A160800, A162795, A169707, A187220, A255366, A256260.

wt[n_] := DigitCount[n, 2, 1];
A151920[n_] := Sum[3^wt[i], {i, 1, n + 1}]/3;
a[n_] := 2*A151920[n - 2] + 2*A151920[n - 3] + n;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 21 2024, after Charlie Neder *)

a(n) = 2*A151920(n) + 2*A151920(n-1) + n + 1. - Charlie Neder, Feb 07 2019

Edited and extended by N. J. A. Sloane, Jan 01 2010

A256263 Triangle read by rows: T(j,k) = 2*k-1 if k is a power of 2, otherwise, between positions that are powers of 2 we have the initial terms of A016969, with j>=0, 1<=k<=A011782(j) and T(0,1) = 0.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5, 11, 17, 15, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 63, 1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89

Offset: 0

Omar E. Pol, Mar 30 2015

Partial sums give A256264.
First differs from A160552 at a(27).
Appears to be a canonical sequence partially related to the cellular automata of A139250, A147562, A162795, A169707, A255366, A256250. See also A256264 and A256260.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
0;
1;
1,3;
1,3,5,7;
1,3,5,7,5,11,17,15;
1,3,5,7,5,11,17,15,5,11,17,23,29,35,41,31;
1,3,5,7,5,11,17,15,5,11,17,23,29,35,41,31,5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,63;
...
Right border gives A000225.
Apart from the initial 0 the row sums give A000302.
Rows converge to A256258.
.
Illustration of initial terms in the fourth quadrant of the square grid:
---------------------------------------------------------------------------
n   a(n)                 Compact diagram
---------------------------------------------------------------------------
0    0     _
1    1    |_|_ _
2    1      |_| |
3    3      |_ _|_ _ _ _
4    1          |_| | | |
5    3          |_ _| | |
6    5          |_ _ _| |
7    7          |_ _ _ _|_ _ _ _ _ _ _ _
8    1                  |_| | | |_ _  | |
9    3                  |_ _| | |_  | | |
10   5                  |_ _ _| | | | | |
11   7                  |_ _ _ _| | | | |
12   5                  | | |_ _ _| | | |
13  11                  | |_ _ _ _ _| | |
14  17                  |_ _ _ _ _ _ _| |
15  15                  |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
16   1                                  |_| | | |_ _  | |_ _ _ _ _ _  | |
17   3                                  |_ _| | |_  | | |_ _ _ _ _  | | |
18   5                                  |_ _ _| | | | | |_ _ _ _  | | | |
19   7                                  |_ _ _ _| | | | |_ _ _  | | | | |
20   5                                  | | |_ _ _| | | |_ _  | | | | | |
21  11                                  | |_ _ _ _ _| | |_  | | | | | | |
22  17                                  |_ _ _ _ _ _ _| | | | | | | | | |
23  15                                  |_ _ _ _ _ _ _ _| | | | | | | | |
24   5                                  | | | | | | |_ _ _| | | | | | | |
25  11                                  | | | | | |_ _ _ _ _| | | | | | |
26  17                                  | | | | |_ _ _ _ _ _ _| | | | | |
27  23                                  | | | |_ _ _ _ _ _ _ _ _| | | | |
28  29                                  | | |_ _ _ _ _ _ _ _ _ _ _| | | |
29  35                                  | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
30  41                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
31  31                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
A256264(n) gives the total number of cells after n-th stage.

Ivan Neretin, Table of n, a(n) for n = 0..8191
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000225, A000302, A011782, A038573, A006257, A016969, A139251, A160552, A256250, A256258, A256260, A256261, A256264, A256265.

Flatten@Join[{0}, NestList[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 6]] (* Ivan Neretin, Feb 14 2017 *)

Terms a(95) to a(98) fixed by Ivan Neretin, Feb 14 2017

A256264 Partial sums of A256263.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 14, 21, 22, 25, 30, 37, 42, 53, 70, 85, 86, 89, 94, 101, 106, 117, 134, 149, 154, 165, 182, 205, 234, 269, 310, 341, 342, 345, 350, 357, 362, 373, 390, 405, 410, 421, 438, 461, 490, 525, 566, 597, 602, 613, 630, 653, 682, 717, 758, 805, 858, 917, 982, 1053, 1130, 1213, 1302, 1365

Offset: 0

Omar E. Pol, Mar 30 2015

First differs from A255747 at a(27).

			Also, written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
0,
1,
2,   5,
6,   9, 14,  21,
22, 25, 30,  37,  42,  53,  70,  85;
86, 89, 94, 101, 106, 117, 134, 149, 154, 165, 182, 205, 234, 269,310,341;
...
It appears that the first column gives 0 together with the terms of A047849, hence the right border gives A002450.
It appears that this triangle at least shares with the triangles from the following sequences; A151920, A255737, A255747, A256249, the positive elements of the columns k, if k is a power of 2.
From _Omar E. Pol, Jan 02 2016: (Start)
Illustration of initial terms in the fourth quadrant of the square grid:
---------------------------------------------------------------------------
n    a(n)                 Compact diagram
---------------------------------------------------------------------------
0     0     _
1     1    |_|_ _
2     2      |_| |
3     5      |_ _|_ _ _ _
4     6          |_| | | |
5     9          |_ _| | |
6    14          |_ _ _| |
7    21          |_ _ _ _|_ _ _ _ _ _ _ _
8    22                  |_| | | |_ _  | |
9    25                  |_ _| | |_  | | |
10   30                  |_ _ _| | | | | |
11   37                  |_ _ _ _| | | | |
12   42                  | | |_ _ _| | | |
13   53                  | |_ _ _ _ _| | |
14   70                  |_ _ _ _ _ _ _| |
15   85                  |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
16   86                                  |_| | | |_ _  | |_ _ _ _ _ _  | |
17   89                                  |_ _| | |_  | | |_ _ _ _ _  | | |
18   94                                  |_ _ _| | | | | |_ _ _ _  | | | |
19  101                                  |_ _ _ _| | | | |_ _ _  | | | | |
20  106                                  | | |_ _ _| | | |_ _  | | | | | |
21  117                                  | |_ _ _ _ _| | |_  | | | | | | |
22  134                                  |_ _ _ _ _ _ _| | | | | | | | | |
23  149                                  |_ _ _ _ _ _ _ _| | | | | | | | |
24  154                                  | | | | | | |_ _ _| | | | | | | |
25  165                                  | | | | | |_ _ _ _ _| | | | | | |
26  182                                  | | | | |_ _ _ _ _ _ _| | | | | |
27  205                                  | | | |_ _ _ _ _ _ _ _ _| | | | |
28  234                                  | | |_ _ _ _ _ _ _ _ _ _ _| | | |
29  269                                  | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
30  310                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
31  341                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the total number of cells in the first n regions of the diagram. A256263(n) gives the number of cells in the n-th region of the diagram.
(End)

Ivan Neretin, Table of n, a(n) for n = 0..8191
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A002450, A011782, A047849, A139250, A151920, A255737, A255747, A256249, A256258, A256260, A256261, A256263, A256265.

Accumulate@Flatten@Join[{0}, NestList[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 5]] (* Ivan Neretin, Feb 14 2017 *)

a(n) = (A256260(n+1) - 1)/4.

A256265 Sums of two successive terms of A256264.

Original entry on oeis.org

1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67, 79, 95, 123, 155, 171, 175, 183, 195, 207, 223, 251, 283, 303, 319, 347, 387, 439, 503, 579, 651, 683, 687, 695, 707, 719, 735, 763, 795, 815, 831, 859, 899, 951, 1015, 1091, 1163, 1199, 1215, 1243, 1283, 1335, 1399, 1475, 1563, 1663, 1775, 1899, 2035, 2183, 2343, 2515, 2667, 2731

Offset: 1

Omar E. Pol, Apr 10 2015

nonn

First differs from A139250 (the toothpick sequence) at a(27).
It appears that both sequences share infinitely many terms, for example: a(1)..a(26), a(31)..a(42), a(47)..a(50), a(63)..a(74), a(79)..a(82), etc.
The main entry for this sequence is A256263.

Cf. A007583, A139250, A255747, A256260, A256261, A256263, A256264.

a(n) = A256264(n) + A256264(n-1).

A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

Original entry on oeis.org

1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 63, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 127, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137

Offset: 1

Omar E. Pol, Apr 04 2015

Row sums give A002001.
The sum of all terms of first n rows gives A000302(n-1).
The rows of triangle A256263 converge to this sequence.
Rows converge to A016969.
First 11 terms agree with A151548.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
3;
5,7;
5,11,17,15;
5,11,17,23,29,35,41,31;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,63;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,107,113,119,125,131,137,143,149,155,161,167,173,179,185,127;
...
Illustration of initial terms in the fourth quadrant of the square grid:
------------------------------------------------------------------------
n   a(n)             Compact diagram
------------------------------------------------------------------------
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1    1      |_| | | |_ _  | |_ _ _ _ _ _  | |
2    3      |_ _| | |_  | | |_ _ _ _ _  | | |
3    5      |_ _ _| | | | | |_ _ _ _  | | | |
4    7      |_ _ _ _| | | | |_ _ _  | | | | |
5    5      | | |_ _ _| | | |_ _  | | | | | |
6   11      | |_ _ _ _ _| | |_  | | | | | | |
7   17      |_ _ _ _ _ _ _| | | | | | | | | |
8   15      |_ _ _ _ _ _ _ _| | | | | | | | |
9    5      | | | | | | |_ _ _| | | | | | | |
10  11      | | | | | |_ _ _ _ _| | | | | | |
11  17      | | | | |_ _ _ _ _ _ _| | | | | |
12  23      | | | |_ _ _ _ _ _ _ _ _| | | | |
13  29      | | |_ _ _ _ _ _ _ _ _ _ _| | | |
14  35      | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
15  41      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
16  31      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
It appears that A241717 can be represented by a similar diagram.

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A000225, A000302, A002001, A011782, A016969, A141548, A241717, A256260, A256261, A256263, A256264.

Nest[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 7] (* Ivan Neretin, Feb 14 2017 *)

A255264 Total number of ON cells in the "Ulam-Warburton" two-dimensional cellular automaton of A147562 after A048645(n) generations.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 85, 89, 101, 149, 341, 345, 357, 405, 597, 1365, 1369, 1381, 1429, 1621, 2389, 5461, 5465, 5477, 5525, 5717, 6485, 9557, 21845, 21849, 21861, 21909, 22101, 22869, 25941, 38229, 87381, 87385, 87397, 87445, 87637

Offset: 1

Omar E. Pol, Feb 19 2015

It appears that these are the terms of A147562, A162795, A169707, A255366, A256250, A256260, whose indices have binary weight 1 or 2.

			Also, written as an irregular triangle in which row lengths are the terms of A028310 the sequence begins:
      1;
      5;
      9,    21;
     25,    37,    85;
     89,   101,   149,   341;
    345,   357,   405,   597,  1365;
   1369,  1381,  1429,  1621,  2389,  5461;
   5465,  5477,  5525,  5717,  6485,  9557, 21845;
  21849, 21861, 21909, 22101, 22869, 25941, 38229, 87381;
  ...
Right border gives the positive terms of A002450.
It appears that the second leading diagonal gives the odd terms of A206374.

Cf. A002450, A028310, A032925, A048645, A075897, A139250, A147562, A162795, A169707, A206374, A209492, A255263, A255366, A256250, A256260.

a(n) = A147562(A048645(n)).
Conjecture 1: a(n) = A162795(A048645(n)).
Conjecture 2: a(n) = A169707(A048645(n)).
Conjecture 3: a(n) = A255366(A048645(n)).
Conjecture 4: a(n) = A256250(A048645(n)).
Conjecture 5: a(n) = A256260(A048645(n)).
a(n) = A032925(A209492(n-1)) (conjectured). - Jon Maiga, Dec 17 2021

A288775 Difference between the total number of toothpicks in the toothpick structure of A139250 that are parallel to the initial toothpick after n odd stages, and the total number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton of A147562.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 4, 4, 28, 0, 0, 0, 4, 0, 4, 4, 28, 0, 4, 4, 28, 4, 28, 32, 132, 0, 0, 0, 4, 0, 4, 4, 28, 0, 4, 4, 28, 4, 28, 32, 132, 0, 4, 4, 28, 4, 28, 32, 132, 4, 28, 32, 132, 32, 136, 176, 524, 0, 0, 0, 4, 0, 4, 4, 28, 0, 4, 4, 28, 4, 28, 32, 132, 0, 4, 4, 28, 4, 28, 32, 132, 4, 28, 32

Offset: 1

Omar E. Pol, Jul 04 2017

It appears that a(n) = 0 if and only if n is a member of A048645.
First differs from A255263 at a(14), with which it shares infinitely many terms.
It appears that A147562(n) = A162795(n) = A169707(n) = A255366(n) = A256250(n) = A256260(n), if n is a member of A048645.

			Written as an irregular triangle T(j,k), k>=1, in which the row lengths are the terms of A011782, the sequence begins:
0;
0;
0,0;
0,0,4,0;
0,0,4,0,4,4,28,0;
0,0,4,0,4,4,28,0,4,4,28,4,28,32,132,0;
0,0,4,0,4,4,28,0,4,4,28,4,28,32,132,0,4,4,28,4,28,32,132,4,28,32,132,32,136,176,524,0;
...
It appears that if k is a power of 2 then T(j,k) = 0.
It appears that every column lists the same terms as its initial term.

Cf. A011782, A048645, A139250, A139251, A147562, A147582, A162793, A162795, A169707, A255263, A255366, A256250, A256260.

a(n) = A162795(n) - A147562(n).

cp's OEIS Frontend

A256261 First differences of A256260.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A160172 T-toothpick sequence (see Comments lines for definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A256263 Triangle read by rows: T(j,k) = 2*k-1 if k is a power of 2, otherwise, between positions that are powers of 2 we have the initial terms of A016969, with j>=0, 1<=k<=A011782(j) and T(0,1) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A256264 Partial sums of A256263.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A256265 Sums of two successive terms of A256264.

Views

Author

Keywords

Comments

Crossrefs

Formula

A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A255264 Total number of ON cells in the "Ulam-Warburton" two-dimensional cellular automaton of A147562 after A048645(n) generations.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A288775 Difference between the total number of toothpicks in the toothpick structure of A139250 that are parallel to the initial toothpick after n odd stages, and the total number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton of A147562.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula