A169707 -id:A169707 - cp's OEIS frontend

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

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

A246335 Second bisection of A246333.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 57, 85, 89, 101, 121, 165, 169, 213, 217, 317, 321, 333, 353, 397, 401, 461, 481, 637, 593, 733, 689, 877, 801, 981, 921, 1157, 1185, 1197, 1217, 1261, 1265, 1325, 1345, 1501, 1457, 1613, 1585, 1829, 1721, 2037, 1913, 2381, 2145, 2477, 2409, 2685

Offset: 0

N. J. A. Sloane, Aug 30 2014

nonn

a(2^k-1) begins 1,5,21,85, which is (4^k-1)/3, but a(15) = 317 not 341, breaking the pattern.

N. J. A. Sloane, Illustration of first 24 generations of A246333
N. J. A. Sloane, Illustration of a(6) = 57 (Note that this is a different configuration of 57 cells from that in generation 6 of A169707, although the configurations of 37 cells at generation 5 are the same up to rotation.)
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Cf. A246333, A246334, A247001. Similar to but different from A169707.

max = 100; (Partition[Total[Flatten[#]]& /@ CellularAutomaton[{493, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, max], 2] /. {a_, b_} -> {a, (max + 1 - Mod[max, 2])^2 - b} )[[All, 2]] (* Jean-François Alcover, Oct 03 2018 *)

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.

A170903 a(n) = 2*A160552(n)-1.

Original entry on oeis.org

1, 1, 5, 1, 5, 9, 13, 1, 5, 9, 13, 9, 21, 33, 29, 1, 5, 9, 13, 9, 21, 33, 29, 9, 21, 33, 37, 41, 77, 97, 61, 1, 5, 9, 13, 9, 21, 33, 29, 9, 21, 33, 37, 41, 77, 97, 61, 9, 21, 33, 37, 41, 77, 97, 69, 41, 77, 105, 117, 161, 253, 257, 125, 1, 5, 9, 13, 9, 21, 33, 29, 9, 21, 33, 37, 41, 77

Offset: 1

Gary W. Adamson, Jan 21 2010

nonn

			When written as a triangle:
1
1, 5;
1, 5, 9, 13;
1, 5, 9, 13, 9, 21, 33, 29;
...
Rows sums are A006516 (this is immediate from the definition).
From _Omar E. Pol_, Feb 17 2015: (Start)
Also, written as an irregular triangle in which the row lengths are the terms of A011782:
1;
1;
5,1;
5,9,13,1;
5,9,13,9,21,33,29,1;
5,9,13,9,21,33,29,9,21,33,37,41,77,97,61,1;
5,9,13,9,21,33,29,9,21,33,37,41,77,97,61,9,21,33,37,41,77,97,69,41,77,105,117,161,253,257,125,1;
Row sums give 1 together with the positive terms of A006516.
It appears that the right border (A000012) gives the smallest difference between A160164 and A169707 in every period.
(End)

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. A006516, A139250, A139251, A160164, A160552, A169707.

It appears that a(n) = A160164(n) - A169707(n). - Omar E. Pol, Feb 17 2015

A169709 Total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 1006".

Original entry on oeis.org

1, 5, 9, 21, 29, 41, 61, 89, 101, 113, 133, 161, 189, 233, 309, 377, 397, 409, 429, 457, 485, 529, 605, 673, 709, 753, 821, 905, 1005, 1169, 1413, 1569, 1605, 1617, 1637, 1665, 1693, 1737, 1813, 1881, 1917, 1961, 2029, 2113, 2213, 2377, 2621, 2777, 2829, 2873, 2941

Offset: 0

N. J. A. Sloane, Apr 17 2010

nonn

Square grid, 4 neighbors per cell, turn ON iff exactly 1, 3 or 4 neighbors are ON; once ON, cells stay ON.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 928.

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

Cf. A139250, A151725, A147562, A147582, A169648, A169649, A169707, A169708, A169710.

Map[Function[Apply[Plus,Flatten[ #1]]], CellularAutomaton[{ 1006, {2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},100]]

A253088 Number of ON cells at generation n of 9-celled totalistic CA defined by Rule 750.

Original entry on oeis.org

1, 9, 12, 40, 24, 88, 48, 148, 80, 164, 172, 296, 236, 380, 324, 500, 436, 552, 552, 644, 688, 936, 788, 1168, 928, 1296, 1044, 1444, 1352, 1768, 1576, 2136, 1728, 2116, 1992, 2264, 2288, 2604, 2536, 3056, 2812, 3380, 3112, 3864, 3484, 4204, 3836, 4764, 4120, 4748, 4456

Offset: 0

N. J. A. Sloane, Feb 12 2015

The keyword "look" refers to the illustration.

N. J. A. Sloane, Illustration of generations 0-15
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Cf. A169707 (5-neighbor analog).

Map[Function[Apply[Plus, Flatten[#1]]],
CellularAutomaton[{750, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, 66]]
ArrayPlot /@ CellularAutomaton[{750, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, 15]

It would be nice to have a recurrence.

A246336 Partial sums of A151548.

Original entry on oeis.org

1, 4, 9, 16, 21, 32, 49, 64, 69, 80, 97, 116, 137, 176, 225, 256, 261, 272, 289, 308, 329, 368, 417, 452, 473, 512, 565, 624, 705, 832, 961, 1024, 1029, 1040, 1057, 1076, 1097, 1136, 1185, 1220, 1241, 1280, 1333, 1392, 1473, 1600, 1729, 1796, 1817, 1856, 1909, 1968, 2049, 2176

Offset: 0

N. J. A. Sloane, Aug 30 2014

Arises in the analysis of a certain 2-D cellular automaton (see A169707).
a(46) = 1729 is also the Hardy-Ramanujan number. - Omar E. Pol, Feb 17 2015
It appears that sums of two successive terms give the numbers greater than 1 in A194811. - Omar E. Pol, Mar 05 2015

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A169707, A151548, A160552, A170903.

G.f.: 1/(1-x^2) + (4*x/(1-x))*mul(1+x^(2^k-1)+2*x^(2^k),k=1..oo).
From Omar E. Pol, Feb 18 2015: (Start)
It appears that:
a(2^k-2) = (2^k-1)^2, if k >= 1.
a(2^k-1) = 4^k, if k >= 1.
a(2^k) = 4^k + 5, if k >= 1.
(End)

A255048 Partial sums of A253088.

Original entry on oeis.org

1, 10, 22, 62, 86, 174, 222, 370, 450, 614, 786, 1082, 1318, 1698, 2022, 2522, 2958, 3510, 4062, 4706, 5394, 6330, 7118, 8286, 9214, 10510, 11554, 12998, 14350, 16118, 17694, 19830, 21558, 23674, 25666, 27930, 30218, 32822, 35358, 38414, 41226, 44606, 47718, 51582, 55066, 59270, 63106, 67870, 71990, 76738, 81194

Offset: 0

Omar E. Pol, Feb 13 2015

Also, total number of cubic ON cells after n generations in a three-dimensional cellular automaton in which A253088(n) gives the number of cubic ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. The structure looks like an irregular stepped pyramid.

Cf. A169707, A253088.

A255263 Differences between the total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 750" using the von Neumann neighborhood and the total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 4, 12, 20, 0, 0, 0, 4, 0, 4, 12, 20, 0, 4, 12, 20, 12, 36, 80, 68, 0, 0, 0, 4, 0, 4, 12, 20, 0, 4, 12, 20, 12, 36, 80, 68, 0, 4, 12, 20, 12, 36, 80, 68, 12, 36, 80, 84, 96, 208, 352, 196, 0, 0, 0, 4, 0, 4, 12, 20, 0, 4, 12, 20, 12, 36, 80, 68, 0, 4, 12, 20, 12, 36, 80, 68, 12, 36, 80

Offset: 1

Omar E. Pol, Feb 19 2015

It appears that the graph of A162795 lies between the graphs of A147562 and A169707.

It appears that a(n) = 0 if and only 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:
0;
0;
0,0;
0,0,4,0;
0,0,4,0,4,12,20,0;
0,0,4,0,4,12,20,0,4,12,20,12,36,80,68,0;
0,0,4,0,4,12,20,0,4,12,20,12,36,80,68,0,4,12,20,12,36,80,68,12,36,80,84,96,208,352,196,0;
...
It appears that if k is a power of 2 then T(j,k) = 0.

Cf. A011782, A048645, A139250, A160164, A162796, A169707, A147562, A255166, A255264.

a(n) = A169707(n) - A162795(n).

cp's OEIS Frontend

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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A246335 Second bisection of A246333.

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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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A170903 a(n) = 2*A160552(n)-1.

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A169709 Total number of ON cells at stage n of two-dimensional cellular automaton defined by "Rule 1006".

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A253088 Number of ON cells at generation n of 9-celled totalistic CA defined by Rule 750.

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A246336 Partial sums of A151548.

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A255048 Partial sums of A253088.

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