A169779 -id:A169779 - cp's OEIS frontend

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

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

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

nonn

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

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 1026 terms from David Applegate and N. J. A. Sloane]
David Applegate, The movie version
David Applegate and N. J. A. Sloane, Table of n, A151724(n), A151723(n) for n = 0..1025
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.]
Bradley Klee, Log-periodic coloring, over the half-hexagon tiling.
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. A147562, A151724, A151725, A161206, A161644, A169779, A169780, A170898, A170905, A182632, A182840, A256536, A256537.

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 *)

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

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

A267700 "Tree" sequence in a 90-degree sector of the cellular automaton of A160720.

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 12, 19, 20, 23, 26, 33, 36, 43, 50, 65, 66, 69, 72, 79, 82, 89, 96, 111, 114, 121, 128, 143, 150, 165, 180, 211, 212, 215, 218, 225, 228, 235, 242, 257, 260, 267, 274, 289, 296, 311, 326, 357, 360, 367, 374, 389, 396, 411, 426, 457, 464, 479, 494, 525, 540, 571, 602, 665, 666, 669, 672, 679, 682, 689

Offset: 0

Omar E. Pol, Jan 19 2016

nonn

Conjecture: this is also the "tree" sequence in a 120-degree sector of the cellular automaton of A266532.
It appears that this is also the partial sums of A038573.
a(n) is also the total number of ON cells after n-th stage in the tree that arises from one of the four spokes in a 90-degree sector of the cellular automaton A160720 on the square grid.
Note that the structure of A160720 is also the "outward" version of the Ulam-Warburton cellular automaton of A147562.
It appears that A038573 gives the number of cells turned ON at n-th stage.
Conjecture: a(n) is also the total number of Y-toothpicks after n-th stage in the tree that arises from one of the three spokes in a 120-degree sector of the cellular automaton of A266532 on the triangular grid.
Note that the structure of A266532 is also the "outward" version of the Y-toothpick cellular automaton of A160120.
It appears that A038573 also gives the number of Y-toothpicks added at n-th stage.
Comment from N. J. A. Sloane, Jan 23 2016: All the above conjectures are true!
From Gus Wiseman, Mar 31 2019: (Start)
a(n) is also the number of nondecreasing binary-containment pairs of positive integers up to n. A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second. For example, the a(1) = 1 through a(6) = 12 pairs are:
  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)
         (2,2)  (1,3)  (1,3)  (1,3)  (1,3)
                (2,2)  (2,2)  (1,5)  (1,5)
                (2,3)  (2,3)  (2,2)  (2,2)
                (3,3)  (3,3)  (2,3)  (2,3)
                       (4,4)  (3,3)  (2,6)
                              (4,4)  (3,3)
                              (4,5)  (4,4)
                              (5,5)  (4,5)
                                     (4,6)
                                     (5,5)
                                     (6,6)
(End)

David Applegate, The movie version
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 toothpick sequences

Cf. A000120, A038573, A147562, A160120, A160720, A161336, A169779, A266532, A266534, A266536.
Cf. A006218, A019565, A070939, A080572, A267610, A267700.
Cf. A325101, A325103, A325104, A325106, A325109, A325110, A325124.

Accumulate[Table[2^DigitCount[n,2,1]-1,{n,0,30}]] (* based on conjecture confirmed by Sloane, Gus Wiseman, Mar 31 2019 *)

a(n) = (A160720(n+1) - 1)/4.
Conjecture 1: a(n) = (A266532(n+1) - 1)/3.
Conjecture 2: a(n) = A160720(n+1) - A266532(n+1).
All of the above conjectures are true. -  N. J. A. Sloane, Jan 23 2016
(Conjecture) a(n) = A267610(n) + n. - Gus Wiseman, Mar 31 2019

A169780 Total number of ON cells after n-th stage in one-sixth slice of hexagonal CA defined in A151723 (including both boundaries).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 185, 203, 211, 225, 249, 277, 297, 329, 361, 363, 367, 373, 383, 393, 403, 421, 447, 465, 475, 493, 523, 561, 595, 637, 695, 729, 737, 751, 775, 803, 831, 875, 943, 1003, 1031

Offset: 0

N. J. A. Sloane, May 11 2010

nonn

Partial sums of A170905.

N. J. A. Sloane, Table of n, a(n) for n = 0..1025
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A151723, A169779, A169781.

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

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.

A255747 Partial sums of A160552.

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, 201, 222, 261, 310, 341, 342, 345, 350, 357, 362, 373, 390, 405, 410, 421, 438, 457, 478, 517, 566, 597, 602, 613, 630, 649, 670, 709, 758, 793, 814, 853, 906, 965, 1046, 1173, 1302, 1365, 1366, 1369, 1374

Offset: 0

Omar E. Pol, Mar 05 2015

nonn

It appears that the sums of two successive terms give the positive terms of the toothpick sequence A139250.
It appears that the odd terms (a bisection) give A162795.
It appears that a(n) is also the total number of ON cells at stage n+1 in one of the four wedges of two-dimensional cellular automaton "Rule 750" using the von Neumann neighborhood (see A169707). Therefore a(n) is also the total number of ON cells at stage n+1 in one of the four quadrants of the NW-NE-SE-SW version of that cellular automaton.
See also the formula section.
First differs from A169779 at a(11).

			Also, written as an irregular triangle in which the row lengths are the terms of A011782 (the number of compositions of n, n >= 0), the sequence begins:
0;
1;
2,   5;
6,   9, 14, 21;
22, 25, 30, 37, 42, 53, 70, 85;
86, 89, 94,101,106,117,134,149,154,165,182,201,222,261,310,341;
...
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 only shares with A151920 the positive elements of the columns 1, 2, 4, 8, 16, ... (the powers of 2).

Ivan Neretin, Table of n, a(n) for n = 0..8191
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.] arXiv:1004.3036
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
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, A151548, A151920, A160552, A162795, A169707, A169779, A246336.

Accumulate[Nest[Join[#, 2 # + Append[Rest@#, 1]] &, {0}, 6]] (* Ivan Neretin, Feb 09 2017 *)

It appears that a(n) + a(n-1) = A139250(n), n >= 1.
It appears that a(2n-1) = A162795(n), n >= 1.
It appears that a(n) = (A169707(n+1) - 1)/4.

A256249 Partial sums of A006257 (Josephus problem).

Original entry on oeis.org

0, 1, 2, 5, 6, 9, 14, 21, 22, 25, 30, 37, 46, 57, 70, 85, 86, 89, 94, 101, 110, 121, 134, 149, 166, 185, 206, 229, 254, 281, 310, 341, 342, 345, 350, 357, 366, 377, 390, 405, 422, 441, 462, 485, 510, 537, 566, 597, 630, 665, 702, 741, 782, 825, 870, 917, 966, 1017, 1070, 1125, 1182, 1241, 1302, 1365, 1366, 1369, 1374

Offset: 0

Omar E. Pol, Mar 20 2015

Also total number of ON states after n generations in one of the four wedges of the one-step rook version (or in one of the four quadrants of the one-step bishop version) of the cellular automaton of A256250.
A006257 gives the number of cells turned ON at n-th stage.
First differs from A255747 at a(11).
First differs from A169779 at a(10).
It appears that the odd terms (a bisection) give A256250.

			Written as an irregular triangle T(n,k), k >= 1, 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, 46, 57, 70, 85;
  86, 89, 94,101,110,121,134,149,166,185,206,229,254,281,310,341;
  ...
Right border, a(2^m-1), gives A002450(m) for m >= 0.
a(2^m-2) = A203241(m) for m >= 2.
It appears that this triangle at least shares with the triangles from the following sequences; A151920, A255737, A255747, the positive elements of the columns k, if k is a power of 2.
From _Omar E. Pol_, Jan 03 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   46                  |_ _ _ _ _| | | |
13   57                  |_ _ _ _ _ _| | |
14   70                  |_ _ _ _ _ _ _| |
15   85                  |_ _ _ _ _ _ _ _|
.
a(n) is also the total number of cells in the first n regions of the diagram. A006257(n) gives the number of cells in the n-th region of the diagram.
(End)

David A. Corneth, Table of n, a(n) for n = 0..9999
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, pp. 37, 41.
Yuri Nikolayevsky and Ioannis Tsartsaflis, Cohomology of N-graded Lie algebras of maximal class over Z_2, arXiv:1512.87676 [math.RA], (2016), page 6.
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, A151920, A169779, A255737, A255747, A256250, A256251, A266540.

(* Based on Birkas Gyorgy's code in A006257 *)
Accumulate[Prepend[Flatten[Table[Range[1,2^n-1,2],{n,0,7}]],0]] (* Ivan N. Ianakiev, Mar 30 2015 *)

a(n)=n++;b=#binary(n>>1);(4^b-1)/3+(n-2^b)^2 \\ David A. Corneth, Mar 21 2015

a(n) = (A256250(n+1) - 1)/4.

A255748 Total number of ON states after n generations of cellular automaton based on triangles in a 60-degree wedge (see Comments lines for definition).

Original entry on oeis.org

1, 3, 4, 8, 11, 13, 14, 22, 29, 35, 40, 44, 47, 49, 50, 66, 81, 95, 108, 120, 131, 141, 150, 158, 165, 171, 176, 180, 183, 185, 186, 218, 249, 279, 308, 336, 363, 389, 414, 438, 461, 483, 504, 524, 543, 561, 578, 594, 609, 623, 636, 648, 659, 669, 678, 686, 693, 699, 704, 708, 711, 713, 714, 778, 841, 903, 964, 1024

Offset: 1

Omar E. Pol, Mar 30 2015

Also partial sums of A080079.
In order to construct the structure we use the following rules:
On the infinite triangular grid we are in a 60-degree wedge with the vertex located on top of the wedge.
The nearest triangular cell to the vertex remains OFF.
At stage 1, we turn ON the cell whose base is adjacent to the previous OFF cell.
At stage n, in the n-th level of the structure, we turn ON k cells connected by their vertices with their bases up, where k = A080079(n).
The cells turned ON remain ON forever.
The structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).
Note that this is also the structure in every one of the six wedges of the structure of A256266.
A080079 gives the number of cells turned ON at n-th stage.

			Illustration of initial terms:
-----------------------------------------------------------
n   A080079(n)   a(n)                  Diagram
-----------------------------------------------------------
.                                        / \
1       1         1                     / T \
2       2         3                    / T T \
3       1         4                   /   T   \
4       4         8                  / T T T T \
5       3        11                 /   T T T   \
6       2        13                /     T T     \
7       1        14               /       T       \
8       8        22              / T T T T T T T T \
9       7        29             /   T T T T T T T   \
10      6        35            /     T T T T T T     \
11      5        40           /       T T T T T       \
12      4        44          /         T T T T         \
13      3        47         /           T T T           \
14      2        49        /             T T             \
15      1        50       /               T               \
...
For n = 15 after 15 generations there are 50 ON cells in the structure, so a(15) = 50.

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

Cf. A047999, A001316, A080079, A139250, A169779, A169788, A170905, A233970, A256256, A256266.

Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 6}] (* Michael De Vlieger, Nov 03 2022 *)

a(n) = A256266(n)/6.

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

Omar E. Pol, Mar 20 2015

nonn

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.

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

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

Omar E. Pol, Apr 02 2015

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.

			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.

David Applegate, The movie version
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

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

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.

A266534 Total number of ON cells after n-th stage in a 90-degree sector of the cellular automaton of A151895.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 13, 16, 21, 24, 29, 36, 37, 40, 43, 46, 53, 58, 65, 74, 83, 96, 107, 120, 133, 136, 143, 150, 157, 168, 179, 190, 209, 226, 247, 258, 271, 286, 299, 314, 327, 334, 349, 364, 381, 406, 417, 434, 455, 470, 493, 514, 533, 562, 583, 608, 631, 646, 661, 680, 703, 736, 761, 782, 807, 836, 857, 892, 927

Offset: 0

Omar E. Pol, Jan 12 2016

nonn

The structure looks like a tree which arises from one of the four spokes of the structure of the cellular automaton of A151895.
a(n) is the total number of ON cells after n-th stage.
For n >> 1 the structure looks like a square which is rotated 45 degrees.
First differs from A161336 (snowflake tree) at a(16).
First differs from A266536 at a(13). - Omar E. Pol, Apr 02 2016

Cf. A151895, A161330, A161336, A169779, A266536, A267700.

a(n) = (A151895(n+1) - 1)/4.

More terms from Omar E. Pol, Apr 02 2016

cp's OEIS Frontend

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

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A267700 "Tree" sequence in a 90-degree sector of the cellular automaton of A160720.

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A169780 Total number of ON cells after n-th stage in one-sixth slice of hexagonal CA defined in A151723 (including both boundaries).

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

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A255747 Partial sums of A160552.

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A256249 Partial sums of A006257 (Josephus problem).

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A255748 Total number of ON states after n generations of cellular automaton based on triangles in a 60-degree wedge (see Comments lines for definition).

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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.

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