A122242 -id:A122242 - cp's OEIS frontend

A179761 Binary expansions of A122242 (A122243) concatenated together to a single binary sequence, so that from each term of A122242, the most significant bits come before the least significant bits.

1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0

Offset: 1

Antti Karttunen, Aug 03 2010

When viewed as a table, the first row contains six terms, the second eight, the third ten, and so on, i.e. row n contains 2(n+2) terms.

A. Karttunen, Table of n, a(n) for n = 1..593664 (containing 768 first rows)

Cf. also A179764 and A179758, A179762 & A179763. A179827-A179828.

A122243 a(n) = A007088(A122242(n)).

Original entry on oeis.org

101010, 11110000, 1110010100, 111010100100, 11101001110000, 1111001010011000, 111100100101110000, 11101100101001011000, 1110110010011011100000, 111100111001000110110000

Offset: 1

Antti Karttunen, Sep 14 2006

A080070, A122230, A122233, A122236, A122240, A122246.

A179771 Quadrisection of the fourth central column of triangle A122242, a(n) = A179770(4*n).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

Conjecture: the last zero (107th) occurs at n=166, after which only ones occur.

A. Karttunen, Table of n, a(n) for n = 1..256

Cf. A179772-A179774, and also A179776, A179831.

a(n) = A179770(4*n).

A179831 Quadrisection of the fourth central column of triangle A122242, a(n) = A179830((4*n)+1).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

Conjecture: the last zero (119th) occurs at n=164, after which only ones occur.

A. Karttunen, Table of n, a(n) for n = 1..256

Cf. A179832-A179834, and also A179771, A179776.

a(n) = A179830((4*n)+1).

A179770 The fourth central column of triangle A122242, i.e., A179761(4), A179761(11), A179761(20), A179761(31), ...

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

A. Karttunen, Table of n, a(n) for n = 1..1024

Cf. A179771-A179774, and also A179775, A179830.

a(n) = A179761(A028875(n+2)).

A376402 Bitwise XOR (centrally aligned) of two consecutive terms of A122242.

Original entry on oeis.org

164, 628, 2444, 10040, 34424, 142400, 612536, 2536016, 8772720, 36320296, 156298040, 648930320, 2246427920, 9290072680, 40123676576, 166398412640, 574717970240, 2376856817864, 10244120543704, 42544644116352, 146496800436256, 607708669110320, 2625008220416552, 10882360875506928, 37586414897168848, 156056124134144296

Offset: 1

Antti Karttunen, Sep 22 2024

Antti Karttunen, Table of n, a(n) for n = 1..1650
Antti Karttunen, Python program for computing this sequence and the associated images.
Antti Karttunen, Terms a(1)-a(2048) drawn as binary strings, in Wolframesque fashion.
Index entries for sequences related to binary expansion of n

Cf. A122242 (A014486, A057548, A082358, A122237, A122241).

Cf. also A327973, A376405, A376412.

A376402(n) = bitxor(A122242(1+n), 2*A122242(n));

Python
```
# See the Links section
```

a(n) = A122242(1+n) XOR 2*A122242(n).

A376412 a(n) = A122242(4+n) XOR 16*A122242(n).

Original entry on oeis.org

14544, 64920, 258096, 925720, 3703264, 16562224, 66158664, 237396008, 948568384, 4239130768, 16949182920, 60767078320, 243080890624, 1085016114240, 4341071150792, 15535530051144, 62225888982288, 277421534227968, 1111070191401136, 3979658311943880, 15908408006551904, 71162952082313488, 284082756324759560, 1019946695587234480

Offset: 1

Antti Karttunen, Sep 22 2024

See comments in A376415.

Antti Karttunen, Table of n, a(n) for n = 1..1650
Antti Karttunen, Python program for computing this sequence and the associated images.
Antti Karttunen, Terms a(1)-a(768) drawn as binary strings, in Wolframesque fashion.
Antti Karttunen, The central parts of terms a(1) - a(20000) drawn in similar fashion. (the widening blank corridor in the middle corresponds with the "central crystallization" attractor present in A122242)
Index entries for sequences related to binary expansion of n

Cf. A122242 (A014486, A057548, A082358, A122237, A122241).

Cf. also A327973, A376402, A376415.

A376412(n) = bitxor(A122242(4+n), 16*A122242(n));

Python
```
# See the Links section
```

a(n) = A122242(4+n) XOR 16*A122242(n).

A080070 Decimal encoding of parenthesizations produced by simple iteration starting from empty parentheses and where each successive parenthesization is obtained from the previous by reflecting it as a general tree/parenthesization, then adding an extra stem below the root and then reflecting the underlying binary tree.

Original entry on oeis.org

0, 10, 1010, 101100, 10110010, 1011100100, 101100110100, 10111001001100, 1011100110100010, 101110011010011000, 10110011101001100010, 1011110010011011000100, 101100111011010001100100

Offset: 0

Antti Karttunen, Jan 27 2003

Corresponding Lisp/Scheme S-expressions are (), (()), (()()), (()(())), (()(())()), (()((())())), (()(())(()())), ...

Conjecture: only the terms in positions 0,1,2 and 4 are symmetric, i.e., A057164(A080068(n)) = A080068(n) (equivalently: A036044(A080069(n)) = A080069(n)) only when n is one of {0,1,2,4}. If this is true, then the formula given in A079438 is exact. I (AK) have checked this up to n=404631 with no other occurrence of a symmetric (general) tree.

			This demonstrates how to get the fourth term 10110010 from the 3rd term 101100. The corresponding binary and general trees plus parenthesizations are shown. The first operation reflects the general tree, the second adds a new stem under the root and the third reflects the underlying binary tree, which induces changes on the corresponding general tree:
..............................................
.....\/................\/\/..........\/\/.....
......\/......\/\/......\/............\/......
.....\/........\/........\/..........\/.......
......(A057164).(A057548)..(A057163)..........
........................o.....................
........................|.....................
........o.....o.........o...o.........o.......
........|.....|..........\./..........|.......
....o...o.....o...o.......o.........o.o.o.....
.....\./.......\./........|..........\|/......
......*.........*.........*...........*.......
..[()(())]..[(())()]..[((())())]..[()(())()]..
...101100....110010....11100100....10110010...

A. Karttunen, Illustration of initial terms
A. Karttunen, Python program for computing this sequence.
A. Karttunen, Terms a(1)-a(256) plotted as a Wolframesque triangle.
A. Karttunen, Terms a(1)-a(512) plotted as a Wolframesque triangle.

Compare to similar Wolframesque plots given in A122229, A122232, A122235, A122239, A122242, A122245. See also A079438, A080067, A080071, A057119.

a(n) = A007088(A080069(n)) = A063171(A080068(n)).

A122245 a(n) = A014486(A122244(n)).

Original entry on oeis.org

44, 232, 920, 3876, 14936, 60568, 248240, 996440, 3876264, 15524272, 63773584, 255477160, 993549616, 3970767760, 16350559552, 65386339632, 254129067336, 1016476056896, 4184726043136, 16740063237448, 65054466609736, 260416091191808

Offset: 1

Antti Karttunen, Sep 14 2006

Questions: to which Wolfram's class does this simple program belong, class 3 or class 4? (Is that classification applicable here? This is not 1D CA, although it may look like one).
Does the "central skyscraper" continue widening forever? (see the image for up to 16384th generation) At what specific points it widens? (A new sequence for that). How does that differ from A122242 and similar sister sequences, with different starting conditions?
Related comments in A179777.

Antti Karttunen, Table of n, a(n) for n = 1..1024
A. Karttunen, Python program for computing this sequence and the associated image.
A. Karttunen, Terms a(1)-a(768) drawn as binary strings, in Wolframesque fashion.
A. Karttunen, Terms a(1)-a(256) drawn as binary strings, showing details better.
Antti Karttunen, The central parts of terms a(1) - a(16384) drawn in similar fashion, showing how the "crystallized central skyscraper" gets slowly wider and wider

A122246 shows the same sequence in binary. Compare to similar Wolframesque plots given in A080070, A122229, A122232, A122235, A122239, A122242, A179755, A179757. Cf. also A179777, A179762, A179417.

A080069 a(n) = A014486(A080068(n)).

Original entry on oeis.org

0, 2, 10, 44, 178, 740, 2868, 11852, 47522, 190104, 735842, 3090116, 11777124, 48557252, 194656036, 778669672, 3117617996, 12677727330, 49850271300, 192901051976, 795560529352, 3243898094388, 12977884832332, 51055591319170

Offset: 0

Antti Karttunen, Jan 27 2003

Note that A080068 can be also obtained as iteration of A072795 o A057506.

Antti Karttunen, Table of n, a(n) for n = 0..512
Antti Karttunen, Python program for computing this sequence and the associated image. (this replaces the old a080069.py.txt)
Antti Karttunen, Terms a(1)-a(512) drawn as binary strings.
Antti Karttunen, Terms a(1)-a(2048) drawn as binary strings.

Same sequence in binary: A080070. Compare with similar Wolframesque plots given in A122229, A122232, A122235, A122239, A122242, A122245, A328111.

Cf. A179758.

Python
```
# See attached program
```

Python program and Wolfram-like plot added by Antti Karttunen, Sep 14 2006

cp's OEIS Frontend

A179761 Binary expansions of A122242 (A122243) concatenated together to a single binary sequence, so that from each term of A122242, the most significant bits come before the least significant bits.

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A122243 a(n) = A007088(A122242(n)).

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A179771 Quadrisection of the fourth central column of triangle A122242, a(n) = A179770(4*n).

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A179831 Quadrisection of the fourth central column of triangle A122242, a(n) = A179830((4*n)+1).

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A179770 The fourth central column of triangle A122242, i.e., A179761(4), A179761(11), A179761(20), A179761(31), ...

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A376402 Bitwise XOR (centrally aligned) of two consecutive terms of A122242.

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PARI

Python

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A376412 a(n) = A122242(4+n) XOR 16*A122242(n).

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PARI

Python

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A122245 a(n) = A014486(A122244(n)).

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A080069 a(n) = A014486(A080068(n)).

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