A256265 -id:A256265 - 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

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.

A256261 First differences of A256260.

Original entry on oeis.org

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.

A266509 Terms of A256263 repeated.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 3, 3, 1, 1, 3, 3, 5, 5, 7, 7, 1, 1, 3, 3, 5, 5, 7, 7, 5, 5, 11, 11, 17, 17, 15, 15, 1, 1, 3, 3, 5, 5, 7, 7, 5, 5, 11, 11, 17, 17, 15, 15, 5, 5, 11, 11, 17, 17, 23, 23, 29, 29, 35, 35, 41, 41, 31, 31, 1, 1, 3, 3, 5, 5, 7, 7, 5, 5, 11, 11, 17, 17, 15, 15, 5, 5, 11, 11, 17, 17, 23, 23, 29, 29

Offset: 1

Omar E. Pol, Jan 02 2016

First differs from A266529 at a(55), with which it shares infinitely many terms.
First differs from A266539 at a(25), with which it shares infinitely many terms.
For an illustration of initial terms consider the diagram of A256263 in the fourth quadrant of the square grid together with a reflected copy in the second quadrant.

			Written as an irregular triangle in which the row lengths are twice the terms of A011782 the sequence begins:
0,0;
1,1;
1,1,3,3;
1,1,3,3,5,5,7,7;
1,1,3,3,5,5,7,7,5,5,11,11,17,17,15,15;
1,1,3,3,5,5,7,7,5,5,11,11,17,17,15,15,5,5,11,11,17,17,23,23,29,29,35,35,41,41,31,31;
...
Row sums give 0 together with A004171.

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

Partial sums give A266510.

Cf. A004171, A011782, A256263, A256264, A256265, A266529, A266539.

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

A266510 Partial sums of A266509.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 7, 10, 11, 12, 15, 18, 23, 28, 35, 42, 43, 44, 47, 50, 55, 60, 67, 74, 79, 84, 95, 106, 123, 140, 155, 170, 171, 172, 175, 178, 183, 188, 195, 202, 207, 212, 223, 234, 251, 268, 283, 298, 303, 308, 319, 330, 347, 364, 387, 410, 439, 468, 503, 538, 579, 620, 651, 682, 683, 684, 687, 690, 695, 700

Offset: 1

Omar E. Pol, Dec 30 2015

nonn

Also A256265 and twice the terms of A256264 interleaved, with a(1) = 0.
It appears that this sequence has a fractal (or fractal-like) behavior.
First differs from A266530 at a(55), with which it shares infinitely many terms.
First differs from A266540 at a(25), with which it shares infinitely many terms.
For an illustration of initial terms consider the diagram of A256264 in the fourth quadrant of the square grid together with a reflected copy in the second quadrant.

Ivan Neretin, Table of n, a(n) for n = 1..8192
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A256263, A256264, A256265, A266509, A266530, A266540.

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

a(2n-1) = A256265(n).
a(2n) = 2 * A256264(n-1).
a(n) = (a(n-1) + a(n+1))/2, if n is an odd number greater than 1.

A266535 Sums of two successive terms of A256249, with a(0) = 0.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67, 83, 103, 127, 155, 171, 175, 183, 195, 211, 231, 255, 283, 315, 351, 391, 435, 483, 535, 591, 651, 683, 687, 695, 707, 723, 743, 767, 795, 827, 863, 903, 947, 995, 1047, 1103, 1163, 1227, 1295, 1367, 1443, 1523, 1607, 1695, 1787, 1883, 1983, 2087, 2195, 2307, 2423, 2543, 2667, 2731

Offset: 0

Omar E. Pol, Jan 02 2016

nonn

Also bisection of A266540.
It appears that this sequence has a fractal-like behavior (see Plot 2, A139250 vs. this sequence).
First differs from both the toothpick sequence A139250 and A256265 at a(12), with which it shares infinitely many terms.

Michel Marcus, 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. A006257, A139250, A256249, A256265, A266539, A266540.

Most@ # + Rest@ # &@ Accumulate@ Join[{0, 0}, Flatten@ Table[Range[1, 2^n - 1, 2], {n, 0, 6}]] (* Michael De Vlieger, Jan 05 2016, after Ivan N. Ianakiev at A256249 *)

f(n)=n++; b=#binary(n>>1); (4^b-1)/3+(n-2^b)^2; \\ A256249
a(n) = if (n, f(n)+f(n-1), 0);

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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A256264 Partial sums of A256263.

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A256261 First differences of A256260.

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A266509 Terms of A256263 repeated.

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A266510 Partial sums of A266509.

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A266535 Sums of two successive terms of A256249, with a(0) = 0.

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