A255366 -id:A255366 - cp's OEIS frontend

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

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

A256260 Total number of ON states after n generations of a cellular automaton-like on the square grid.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 57, 85, 89, 101, 121, 149, 169, 213, 281, 341, 345, 357, 377, 405, 425, 469, 537, 597, 617, 661, 729, 821, 937, 1077, 1241, 1365, 1369, 1381, 1401, 1429, 1449, 1493, 1561, 1621, 1641, 1685, 1753, 1845, 1961, 2101, 2265, 2389, 2409, 2453, 2521, 2613, 2729, 2869, 3033, 3221, 3433, 3669, 3929, 4213, 4521, 4853, 5209, 5461

Offset: 1

Omar E. Pol, Mar 28 2015

First differs from A169707 at a(28).
Compare A169707. It appears that both sequences share infinitely many terms, for example: a(1)..a(27), a(31)..a(43), a(47)..a(51), etc.
See also the conjecture in the Example section.
The main entry for this sequence is A256263.
A256261 gives the number of cells turned ON at n-th stage.

			Written as an irregular triangle T(j,k), k>=1, in which the row lengths are the terms of A011782, the sequence begins:
1;
5;
9,   21;
25,  37, 57, 85;
89, 101,121,149,169,213,281,341;
345,357,377,405,425,469,537,597,617,661,729,821,937,1077,1241,1365;
...
The right border gives the positive terms of A002450.
It appears that this triangle at least shares with the triangles from the following sequences; A147562, A162795, A169707, A255366, A256250, the positive elements of the columns k, if k is a power of 2.

Cf. A002450, A139250, A147562, A162795, A169707, A255264, A255366, A256250, A256258, A256261, A256263, A256264, A256265.

a(n) = 1 + 4*A256264(n-1).

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

A256250 Total number of ON states after n generations of a cellular automaton on the square grid.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 57, 85, 89, 101, 121, 149, 185, 229, 281, 341, 345, 357, 377, 405, 441, 485, 537, 597, 665, 741, 825, 917, 1017, 1125, 1241, 1365, 1369, 1381, 1401, 1429, 1465, 1509, 1561, 1621, 1689, 1765, 1849, 1941, 2041, 2149, 2265, 2389, 2521, 2661, 2809, 2965, 3129, 3301, 3481, 3669, 3865, 4069, 4281, 4501, 4729, 4965, 5209, 5461

Offset: 1

Omar E. Pol, Mar 20 2015

A256251 gives the number of cells turned ON at n-th stage.
Note that the number of cells turned ON at n-th stage in each one of its four quadrants is also A006257 (Josephus problem). For more information see A256249.
It appears that this is also a bisection of A256249.
First differs from A169707 at a(13), but both sequences share infinitely many terms. This one is simpler. Compare A169707.

			Also, written as an irregular triangle T(n,k), k >= 1, in which the row lengths are the terms of A011782 the sequence begins:
1;
5;
9,   21;
25,  37, 57, 85;
89, 101,121,149,185,229,281,341;
345,357,377,405,441,485,537,597,665,741,825,917,1017,1125,1241,1365;
...
Right border gives the positive terms of A002450.
It appears that this triangle at least shares with the triangles from the following sequences; A147562, A162795, A169707, A255366, the positive elements of the columns k, if k is a power of 2.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 37.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to the Josephus Problem

Cf. A002450, A006257, A011782, A139250, A147562, A160720, A162795, A169707, A246335, A255366, A256138, A256249, A256251.

1 + 4*Accumulate@ Prepend[Flatten@ Table[Range[1, 2^n - 1, 2], {n, 0, 7}], 0] (* Michael De Vlieger, Nov 03 2022, after Ivan N. Ianakiev at A256249 *)

a(n) = 1 + 4*A256249(n-1), n >= 1.

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

A262609 Divisors of 1728.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728

Offset: 1

Omar E. Pol, Nov 20 2015

A000578(12) = 1728 is the cube of 12.
The number of divisors of 1728 is A000005(1728) = 28.
The sum of the divisors of 1728 is A000203(1728) = 5080.
The prime factorization of 1728 is 2^6 * 3^3.
1728 + 1 = A001235(1) = A011541(2) = 1729 is the Hardy-Ramanujan number.
Three examples related to cellular automata:
1728 is also the number of ON cells after 32 generations of the cellular automata A160239 and A253088.
1728 is also the total number of ON cells around the central ON cell after 24 generations of the cellular automata A160414 and A256530.
1728 is also the total number of ON cells around the central ON cell after 43 generations of the cellular automata A160172 and A255366.

			a(3) * a(26) = 3 * 576 = 1728.
a(4) * a(25) = 4 * 432 = 1728.
a(5) * a(24) = 6 * 288 = 1728.

N. J. A. Sloane, Illustration of A160239(32) = A246030(5) = 1728
Index entries for sequences related to divisors of numbers

Cf. A000005, A000203, A000578, A001235, A011541, A133029, A160172, A160239, A160414, A246030, A253088, A255366, A256530.

Mathematica
```
Divisors[1728]
```
PARI
```
divisors(1728)
```
Sage
```
divisors(1728);
```

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

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

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A256260 Total number of ON states after n generations of a cellular automaton-like on the square grid.

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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.

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A256250 Total number of ON states after n generations of a cellular automaton on the square grid.

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

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A262609 Divisors of 1728.

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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.

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