A256249 -id:A256249 - cp's OEIS frontend

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

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);

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.

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.

A266540 Partial sums of A266539.

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, 83, 92, 103, 114, 127, 140, 155, 170, 171, 172, 175, 178, 183, 188, 195, 202, 211, 220, 231, 242, 255, 268, 283, 298, 315, 332, 351, 370, 391, 412, 435, 458, 483, 508, 535, 562, 591, 620, 651, 682, 683, 684, 687, 690, 695, 700

Offset: 1

Omar E. Pol, Jan 02 2016

nonn

Also A266535 and twice the terms of A256249 interleaved, or in other words A266535 and A266538 interleaved.
It appears that this sequence has a fractal (or fractal-like) behavior.
First differs from both A266510 and A266530 at a(25), with which it shares infinitely many terms.
For an illustration of initial terms consider the diagram of A256249 in the fourth quadrant of the square grid together with a reflected copy in the second quadrant.
Also the third sequence of Betti numbers of the Lie algebra m_0(n) over Z_2. See the Nikolayevsky-Tsartsaflis paper, pages 2 and 6. Note that a(n) is denoted by b_3(m_0(n)).

Robert Israel, Table of n, a(n) for n = 1..10000
Yuri Nikolayevsky and Ioannis Tsartsaflis, Cohomology of N-graded Lie algebras of maximal class over Z_2, arXiv:1512.87676 [math.RA], (2016); see pp. 2 and 6.
Index entries for sequences related to the Josephus Problem

Cf. A006257 (Josephus problem), A256249, A266535, A266510, A266530, A266538, A266539.

A006257[0]:=0: for n from 1 to 100 do A006257[n]:=(A006257[n-1]+1) mod n +1: end do:
ListTools:-PartialSums([seq(A006257[i]$2,i=0..100)]); # Robert Israel, Jan 13 2016

Join[{0, 0}, Table[{k, k}, {n, 1, 6}, {k, 1, 2^n-1, 2}] // Flatten] // Accumulate (* Jean-François Alcover, Sep 19 2018 *)

a(2n-1) = A266535(n).
a(2n) = 2 * A256249(n-1) = A266538(n-1).
a(n) = (a(n-1) + a(n+1))/2, if n is an odd number greater than 1.
G.f.: (x^3+x^5)/(1-2*x+2*x^3-x^4) - x*(1-x)^(-2)*Sum_{k>=1} 2^k*x^(2^(1+k)). - Robert Israel, Jan 13 2016

A266539 Terms of A006257 (Josephus problem) 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, 9, 9, 11, 11, 13, 13, 15, 15, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25

Offset: 1

Omar E. Pol, Jan 02 2016

First differs from both A266509 and A266529 at a(25), and shares with them infinitely many terms.

			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, 9, 9, 11, 11, 13, 13, 15, 15;
   ...
Row sums give 0 together with A004171.

Partial sums give A266540.

Cf. A004171, A006257, A011782, A256249, A266509, A266529, A266535.

A006257[0]:=0: for n from 1 to 100 do A006257[n]:=(A006257[n-1]+1) mod n +1: end do:
seq(A006257[i]$2,i=0..100); # Robert Israel, Jan 13 2016

Join[{0, 0}, Table[SeriesCoefficient[(1 + x^2)/((-1 + x)^2 (1 + x)), {x, 0, m}], {n, 6}, {m, 0, 2^n - 1}]] // Flatten (* Michael De Vlieger, Jan 05 2016 *)

G.f.: (x^2 + x^4)/(1 - x - x^2 + x^3) - (1 - x)^(-1)*Sum_{k>=1} 2^k*x^(2^(k+1)). - Robert Israel, Jan 13 2016

Offset changed to 1 by Ivan Neretin, Feb 09 2017

A266538 Twice the partial sums of A006257 (Josephus problem).

Original entry on oeis.org

0, 2, 4, 10, 12, 18, 28, 42, 44, 50, 60, 74, 92, 114, 140, 170, 172, 178, 188, 202, 220, 242, 268, 298, 332, 370, 412, 458, 508, 562, 620, 682, 684, 690, 700, 714, 732, 754, 780, 810, 844, 882, 924, 970, 1020, 1074, 1132, 1194, 1260, 1330, 1404, 1482, 1564, 1650, 1740, 1834, 1932, 2034, 2140, 2250, 2364, 2482, 2604

Offset: 0

Omar E. Pol, Jan 12 2016

Index entries for sequences related to the Josephus Problem.

Bisection of A266540.

Cf. A006257, A256249, A266535, A266540.

2*Accumulate[Flatten[{0, Table[Range[1, 2^n - 1, 2], {n, 0, 7}]}]] (* Jake L Lande, Aug 05 2024 *)

a(n) = 2 * A256249(n).

cp's OEIS Frontend

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

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A256264 Partial sums of A256263.

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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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A266540 Partial sums of A266539.

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A266539 Terms of A006257 (Josephus problem) repeated.

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A266538 Twice the partial sums of A006257 (Josephus problem).

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