A139251 -id:A139251 - cp's OEIS frontend

A161415 First differences of A160414.

1, 8, 12, 28, 12, 36, 36, 92, 12, 36, 36, 108, 36, 108, 108, 292, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 908, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 36, 108, 108, 324, 108, 324, 324, 972, 108, 324, 324

Offset: 1

Omar E. Pol, May 20 2009, Jun 13 2009

nonn

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
Omar E. Pol, Illustration of initial terms [From _Omar E. Pol_, Nov 11 2009]
D. Applegate, Omar E. Pol, N. J. A. Sloane, The toothpick sequence and other sequences from cellular automata, arXiv:1004.3036 [math.CO] [From _R. J. Mathar_, Oct 16 2010]

Cf. A139250, A139251, A160411, A160413, A160414, A160417.
Cf. A160727.
Cf. A048883, A161411, A162349. [From Omar E. Pol, Nov 11 2009]

Contribution from R. J. Mathar, Oct 16 2010: (Start)
isA000079 := proc(n) if type(n,'even') then nops(numtheory[factorset](n)) = 1 ; else false ; fi ; end proc:
A048883 := proc(n) 3^wt(n) ; end proc:
A161415 := proc(n) if n = 1 then 1; elif isA000079(n) then 4*A048883(n-1)-2*n ; else 4*A048883(n-1) ; end if; end proc: seq(A161415(n),n=1..90) ; (End)

a[1] = 1; a[n_] := 4*3^DigitCount[n-1, 2, 1] - If[IntegerQ[Log[2, n]], 2n, 0];
Array[a, 60] (* Jean-François Alcover, Nov 17 2017, after N. J. A. Sloane *)

For n > 1, a(n) = 4*A048883(n-1), except a(n) = 4*A048883(n-1) - 2n if n is a power of 2. - N. J. A. Sloane, Jul 13 2009

More terms from R. J. Mathar, Oct 16 2010

A162797 a(n) = difference between the number of toothpicks of A139250 that are orthogonal to the initial toothpick and the number of toothpicks that are parallel to the initial toothpick, after n even rounds.

Original entry on oeis.org

1, 1, 5, 1, 5, 5, 17, 1, 5, 5, 17, 5, 17, 21, 49, 1, 5, 5, 17, 5, 17, 21, 49, 5, 17, 21, 49, 21, 53, 81, 129, 1, 5, 5, 17, 5, 17, 21, 49, 5, 17, 21, 49, 21, 53, 81, 129, 5, 17, 21, 49, 21, 53, 81, 129

Offset: 1

Omar E. Pol, Jul 14 2009

nonn

It appears that a(2^k) = 1, for k >= 0. [From Omar E. Pol, Feb 22 2010]

			Contribution from _Omar E. Pol_, Feb 22 2010: (Start)
If written as a triangle:
1;
1,5;
1,5,5,17;
1,5,5,17,5,17,21,49;
1,5,5,17,5,17,21,49,5,17,21,49,21,53,81,129;
1,5,5,17,5,17,21,49,5,17,21,49,21,53,81,129,5,17,21...
Rows converge to A173464.
(End)
Contribution from Omar E. Pol, Apr 01 2011 (Start):
It appears that the final terms of rows give A000337.
It appears that row sums give A006516.
(End)

Nathaniel Johnston, Table of n, a(n) for n = 1..94
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, A159791, A159792, A162793, A162794, A162795, A162796.

Cf. A000337, A058922, A173464. [From Omar E. Pol, Feb 22 2010]

a(n) = A162796(n) - A162795(n).

Edited by Omar E. Pol, Jul 18 2009
More terms from Omar E. Pol, Feb 22 2010
More terms (a(51)-a(55)) from Nathaniel Johnston, Mar 30 2011

A169708 First differences of A169707.

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, 76, 84, 156, 196, 124, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 76, 84, 156, 196, 124, 20, 44, 68, 76, 84, 156, 196, 140, 84, 156, 212, 236, 324, 508, 516, 252, 4, 12, 20, 28, 20

Offset: 0

N. J. A. Sloane, Apr 17 2010

			From _Omar E. Pol_, Feb 13 2015: (Start)
Written as an irregular triangle in which row lengths are 1,1,2,4,8,16,32,... the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,28,20,44,68,60;
4,12,20,28,20,44,68,60,20,44,68,76,84,156,196,124;
4,12,20,28,20,44,68,60,20,44,68,76,84,156,196,124,20,44,68,76,84,156,196,140,84,156,212,236,324,508,516,252;
It appears that the row sums give A000302.
It appears that the right border gives A173033.
(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

Cf. A000302, A011782, A139251, A151548, A152980, A153006, A160164, A160552, A169707 (partial sums), A170903, A173033, A253088.

It appears that a(n) = 4*A160552(n), n >= 1. - Omar E. Pol, Feb 13 2015

Initial 1 added by Omar E. Pol, Feb 13 2015

A182841 Number of toothpicks added at n-th stage in the toothpick structure of A182840.

Original entry on oeis.org

0, 1, 4, 8, 14, 16, 14, 24, 38, 32, 14, 24, 46, 64, 54, 56, 86, 64, 14, 24, 46, 64, 62, 80, 126, 144, 86, 56, 110, 168, 158, 144, 198, 128, 14, 24, 46, 64, 62, 80, 126, 144, 94, 80, 142, 224, 238, 224, 286, 304, 150, 56, 110, 168, 182, 216, 326, 408, 302, 176, 262, 408, 414, 360, 438, 256

Offset: 0

Omar E. Pol, Dec 09 2010

First differences of A182840.

			From _Omar E. Pol_, Nov 01 2014: (Start)
When written as an irregular triangle:
0;
1;
4;
8;
14, 16;
14, 24, 38, 32;
14, 24, 46, 64, 54, 56, 86, 64;
14, 24, 46, 64, 62, 80, 126, 144, 86, 56, 110, 168, 158, 144, 198, 128;
14, 24, 46, 64, 62, 80, 126, 144, 94, 80, 142, 224, 238, 224, 286, 304, 150, 56, 110, 168, 182, 216, 326, 408, 302, 176, 262, 408, 414, 360, 438, 256;
...
(End)

Olaf Voß, Table of n, a(n) for n = 0..1000
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
Olaf Voß, Toothpick structures on hexagonal net
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A139250, A139251, A160121, A161207, A182633, A182635, A151724, A182842.

a(2^k + 1) = 2^(k+2), at least for 0 <= k <= 9. - Omar E. Pol, Nov 01 2014

More terms from Olaf Voß, Dec 24 2010

Wiki link added by Olaf Voß, Jan 14 2011

A194445 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194444.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 11, 8, 4, 8, 16, 24, 12, 20, 25, 16, 4, 8, 16, 24, 28, 36, 42, 44, 20, 24, 40, 64, 32, 44, 53, 32, 4, 8, 16, 24, 28, 36, 44, 52, 42, 48, 60, 100, 68, 84, 83, 84, 28, 24, 44, 72, 84, 104, 116, 132, 54, 56, 92, 144, 72, 92, 109, 64, 4

Offset: 0

Omar E. Pol, Aug 24 2011

Essentially the first differences of A194444.

First differs from A220525 at a(13). - Omar E. Pol, Mar 23 2013

			Contribution from _Omar E. Pol_, Dec 05 2012 (Start):
Triangle begins:
0;
1;
2;
4,4;
4,8,11,8;
4,8,16,24,12,20,25,16;
4,8,16,24,28,36,42,44,20,24,40,64,32,44,53,32;
(End)

Row lengths give 1 together with A011782. Right border gives 0 together with A000079.

Cf. A139250, A139251, A172311, A182635, A182839, A194271, A194433, A194435, A194441, A194443, A194444, A212009.

It appears that a(2^k+1) = 4, if k >= 1.

a(n) = A194435(n)/4. - Omar E. Pol, Mar 23 2013

More terms from Omar E. Pol, Mar 23 2013

A228367 n-th element of the ruler function plus the highest power of 2 dividing n.

Original entry on oeis.org

2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 38, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 71, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7

Offset: 1

Omar E. Pol, Aug 22 2013

a(n) is also the length of the n-th pair of orthogonal line segments in a diagram of compositions, see example.
a(n) is also the largest part plus the number of parts of the n-th region of the mentioned diagram (if the axes both "x" and "y" are included in the diagram).
a(n) is also the number of toothpicks added at n-th stage to the structure of A228366. Essentially the first differences of A228366.
The equivalent sequence for partitions is A207779.

			Illustration of initial terms (n = 1..16) using a diagram of compositions in which A001511(n) is the length of the horizontal line segment in row n and A006519(n) is the length of the vertical line segment ending in row n. Hence a(n) is the length of the n-th pair of orthogonal line segments. Also counting both the x-axis and the y-axis we have that A001511(n) is also the largest part of the n-th region of the diagram and A006519(n) is also the number of parts of the n-th region of the diagram, see below.
---------------------------------------------------------
.                Diagram of
n   A001511(n)  compositions   A006519(n)    a(n)
---------------------------------------------------------
1       1        _| | | | |        1          2
2       2        _ _| | | |        2          4
3       1        _|   | | |        1          2
4       3        _ _ _| | |        4          7
5       1        _| |   | |        1          2
6       2        _ _|   | |        2          4
7       1        _|     | |        1          2
8       4        _ _ _ _| |        8         12
9       1        _| | |   |        1          2
10      2        _ _| |   |        2          4
11      1        _|   |   |        1          2
12      3        _ _ _|   |        4          7
13      1        _| |     |        1          2
14      2        _ _|     |        2          4
15      1        _|       |        1          2
16      5        _ _ _ _ _|       16         21
...
If written as an irregular triangle the sequence begins:
  2;
  4;
  2, 7;
  2, 4, 2, 12;
  2, 4, 2, 7, 2, 4, 2, 21;
  2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 38;
  ...
Row lengths is A011782. Right border gives A005126.
Counting both the x-axis and the y-axis we have that A038712(n) is the area (or the number of cells) of the n-th region of the diagram. Note that adding only the x-axis to the diagram we have a tree. - _Omar E. Pol_, Nov 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..16383
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A001511, A001792, A005126, A006519, A011782, A038712, A139250, A139251, A207779, A228366.

Array[1 + # + 2^# &[IntegerExponent[#, 2]] &, 84] (* Michael De Vlieger, Nov 06 2018 *)

A228367(n) = (1 + valuation(n,2) + 2^valuation(n,2)); \\ Antti Karttunen, Nov 06 2018

def A228367(n): return (m:=n&-n)+m.bit_length() # Chai Wah Wu, Jul 14 2022

a(n) = A001511(n) + A006519(n).

A289841 Number of elements added at n-th stage to the structure of the complex square cross described in A289840.

Original entry on oeis.org

0, 1, 2, 8, 8, 8, 8, 32, 16, 16, 16, 48, 16, 16, 16, 64, 48, 32, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48, 48, 32, 80, 16, 16, 16, 64, 48

Offset: 0

Omar E. Pol, Jul 14 2017

For n = 0..17 the sequence is similar to the known toothpick sequences.
The surprising fact is that for n >= 18 the sequence has a periodic tail. More precisely, it has period 8: repeat [32, 80, 16, 16, 16, 64, 48, 48]. This tail is in accordance with the expansion of the four arms of the cross. The tail also can be written starting from the 20th stage, with period 8: repeat [16, 16, 16, 64, 48, 48, 32, 80], (see example).
This sequence is essentially the first differences of A289840. The behavior is similar to A290221 and A294021 in the sense that these three sequences from cellular automata have the property that after the initial terms the continuation is a periodic sequence. - Omar E. Pol, Oct 29 2017

			For n = 0..17 the sequence is 0, 1, 2, 8, 8, 8, 8, 32, 16, 16, 16, 48, 16, 16, 16, 64, 48, 32;
Terms 18 and beyond can be arranged in a rectangular array with eight columns as shown below:
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
32, 80, 16, 16, 16, 64, 48, 48;
...
On the other hand, in accordance with the periodic structure of the arms of the square cross, the terms 20 and beyond can be arranged in a rectangular array with eight columns as shown below:
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
16, 16, 16, 64, 48, 48, 32, 80;
...

Colin Barker, Table of n, a(n) for n = 0..1000
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 linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A004767, A008584, A042963, A139250, A139251, A220500, A220501, A289840, A290221, A294021.

concat(0, Vec(x*(1 + 2*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 32*x^6 + 16*x^7 + 15*x^8 + 14*x^9 + 40*x^10 + 8*x^11 + 8*x^12 + 8*x^13 + 32*x^14 + 32*x^15 + 16*x^16 + 16*x^17 + 32*x^18 + 16*x^24) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)) + O(x^100))) \\ Colin Barker, Nov 12 2017

G.f.: x*(1 + 2*x + 8*x^2 + 8*x^3 + 8*x^4 + 8*x^5 + 32*x^6 + 16*x^7 + 15*x^8 + 14*x^9 + 40*x^10 + 8*x^11 + 8*x^12 + 8*x^13 + 32*x^14 + 32*x^15 + 16*x^16 + 16*x^17 + 32*x^18 + 16*x^24) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)). - Colin Barker, Nov 12 2017

A160411 Number of cells turned "ON" at n-th stage of A160117.

Original entry on oeis.org

1, 8, 4, 28, 8, 52, 12, 76, 16, 100, 20, 124, 24, 148, 28, 172, 32, 196, 36, 220, 40, 244, 44, 268, 48, 292, 52, 316, 56, 340, 60, 364, 64, 388, 68, 412, 72, 436, 76, 460, 80, 484, 84, 508, 88, 532, 92, 556, 96

Offset: 1

Omar E. Pol, Jun 13 2009

First differences of A160117.

It appears that one of the bisections is A008574. - Omar E. Pol, Sep 20 2011

Paolo Xausa, Table of n, a(n) for n = 1..10000
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 linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A139251, A160117, A160411, A160413, A160415.

LinearRecurrence[{0,2,0,-1},{1,8,4,28,8,52},100] (* Paolo Xausa, Sep 01 2023 *)

G.f.: x*(x^2+1)*(4*x^3 + x^2 + 8*x + 1) / ((x-1)^2*(x+1)^2). - Colin Barker, Mar 04 2013
From Colin Barker, Apr 06 2013: (Start)
a(n) = -11 - 9*(-1)^n + (7 + 5*(-1)^n)*n for n > 2.
a(n) = 2*a(n-2) - a(n-4) for n > 6. (End)

a(10)-a(27) from Omar E. Pol, Mar 26 2011

More terms from Colin Barker, Apr 06 2013

A160412 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 3, 12, 21, 48, 57, 84, 111, 192, 201, 228, 255, 336, 363, 444, 525, 768, 777, 804, 831, 912, 939, 1020, 1101, 1344, 1371, 1452, 1533, 1776, 1857, 2100, 2343, 3072, 3081, 3108, 3135, 3216, 3243, 3324, 3405, 3648, 3675, 3756, 3837, 4080, 4161, 4404, 4647

Offset: 0

Omar E. Pol, May 20 2009, Jun 01 2009

nonn

From Omar E. Pol, Nov 10 2009: (Start)
On the infinite square grid, consider the outside corner of an infinite square.
We start at round 0 with all cells in the OFF state.
The rule: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.
At round 1, we turn ON three cells around the corner of the infinite square, forming a concave-convex hexagon with three exposed vertices.
At round 2, we turn ON nine cells around the hexagon.
At round 3, we turn ON nine other cells. Three cells around of every corner of the hexagon.
And so on.
Shows a fractal-like behavior similar to the toothpick sequence A153006.
For the first differences see the entry A162349.
For more information see A160410, which is the main entry for this sequence.
(End)

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
...77..77..77..77
...766667..766667
....6556....6556.
....654444444456.
...76643344334667
...77.43222234.77
......44211244...
00000000001244...
00000000002234.77
00000000004334667
0000000000444456.
0000000000..6556.
0000000000.766667
0000000000.77..77
0000000000.......
0000000000.......
0000000000.......

Michael De Vlieger, Table of n, a(n) for n = 0..1000
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.]
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. 31.
Omar E. Pol, Illustration of initial terms [From _Omar E. Pol_, Nov 10 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata - _Omar E. Pol_, Nov 10 2009

Cf. A139250, A139251, A153006, A152980, A160410, A160414.

Cf. A130665, A162349. - Omar E. Pol, Nov 10 2009

a[n_] := 3*Sum[3^DigitCount[k, 2, 1], {k, 0, n - 1}]; Array[a, 48, 0] (* Michael De Vlieger, Nov 01 2022 *)

From Omar E. Pol, Nov 10 2009: (Start)
a(n) = A160410(n)*3/4.
a(0) = 0, a(n) = A130665(n-1)*3, for n>0.
(End)

More terms from Omar E. Pol, Nov 10 2009
Edited by Omar E. Pol, Nov 11 2009
More terms from Nathaniel Johnston, Nov 06 2010
More terms from Colin Barker, Apr 19 2015

A160422 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose virtual skeleton is a polyedge as the toothpick structure of A139250 but with toothpicks of length 6.

Original entry on oeis.org

0, 7, 19, 41, 63, 87, 131, 193, 235, 259, 303, 367, 435, 527, 675, 837, 919, 943, 987, 1051, 1119, 1211, 1359, 1523, 1631, 1723, 1875, 2071, 2299, 2631, 3087, 3489, 3651, 3675, 3719, 3783, 3851, 3943, 4091, 4255, 4363, 4455, 4607, 4803, 5031, 5363, 5819, 6223, 6411

Offset: 0

Omar E. Pol, May 20 2009

nonn

a(n) is also the number of grid points that are covered after n-th stage by an polyedge as the toothpick structure of A139250, but with toothpicks of length 6.

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, A147614, A160118, A160120, A160170, A160420, A160430.

a(n) = A147614(n)+4*A139250(n) = A160420(n)+2*A139250(n) since each toothpick covers exactly four more grid points than the corresponding toothpick in A147614.

More terms and formula from Nathaniel Johnston, Nov 13 2010

cp's OEIS Frontend

A161415 First differences of A160414.

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A162797 a(n) = difference between the number of toothpicks of A139250 that are orthogonal to the initial toothpick and the number of toothpicks that are parallel to the initial toothpick, after n even rounds.

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A169708 First differences of A169707.

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A182841 Number of toothpicks added at n-th stage in the toothpick structure of A182840.

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A194445 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194444.

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A228367 n-th element of the ruler function plus the highest power of 2 dividing n.

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A289841 Number of elements added at n-th stage to the structure of the complex square cross described in A289840.

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A160411 Number of cells turned "ON" at n-th stage of A160117.

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