A122245 -id:A122245 - cp's OEIS frontend

A179762 Binary expansions of A122245 (A122246) concatenated together to a single binary sequence, so that from each term of A122245, the most significant bits come before the least significant bits.

1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 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 A179765 and A179758, A179761 & A179763. A179827-A179829.

A122246 a(n) = A007088(A122245(n)).

Original entry on oeis.org

101100, 11101000, 1110011000, 111100100100, 11101001011000, 1110110010011000, 111100100110110000, 11110011010001011000, 1110110010010110101000, 111011001110000110110000

Offset: 1

Antti Karttunen, Sep 14 2006

A080070, A122230, A122233, A122236, A122240, A122243.

A179776 Quadrisection of the fourth central column of triangle A122245, a(n) = A179775((4*n)-2).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

Conjecture: the last zero (108th) occurs at n=163, after which only ones occur.

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

Cf. A179777-A179779, and also A179771, A179831.

a(n) = A179775((4*n)-2).

A179775 The fourth central column of triangle A122245, i.e., A179762(4), A179762(11), A179762(20), A179762(31), ...

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

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

Cf. A179776-A179779, and also A179770, A179830.

a(n) = A179762(A028875(n+2)).

A376405 Bitwise XOR (centrally aligned) of two consecutive terms of A122245.

Original entry on oeis.org

176, 584, 2068, 9232, 38952, 135296, 567096, 2444568, 10136288, 34920688, 144732808, 625792608, 2598216176, 8989149792, 37119010736, 160422522664, 665656629200, 2297815400576, 9505629937992, 41066855413976, 169932530966160, 589165636912400, 2439104800321640, 10514745879265952, 43543845360254320, 149771860125187648

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. A122245 (A014486, A057548, A082358, A122237, A122244).

Cf. also A327973, A376402, A376415.

A376405(n) = bitxor(A122245(1+n), 2*A122245(n));

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

a(n) = A122245(1+n) XOR 2*A122245(n).

A376415 a(n) = A122245(4+n) XOR 16*A122245(n).

Original entry on oeis.org

14488, 57880, 258096, 1033752, 3702824, 14821424, 66158736, 264831016, 948570032, 3798139024, 16949183552, 67829237680, 243080889928, 972279514176, 4341071097344, 17360471721544, 62225889019592, 248568875068928, 1111070190653712, 4438793067349704, 15908408008868528, 63634253845942544, 284082756299099488, 1136310075423425200

Offset: 1

Antti Karttunen, Sep 22 2024

This seems to preserve more of the "wavy texture" present in A122245 than what A376412 does vis-a-vis A122242. Compare the corresponding illustrations.

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 region in the middle corresponds with the "central crystallization" attractor present in A122245)
Index entries for sequences related to binary expansion of n

Cf. A122245 (A014486, A057548, A082358, A122237, A122244).

Cf. also A327973, A376405, A376412.

A376415(n) = bitxor(A122245(4+n), 16*A122245(n));

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

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)).

A122242 a(n) = A014486(A122241(n)).

Original entry on oeis.org

42, 240, 916, 3748, 14960, 62104, 248176, 969304, 3876576, 15962544, 63772488, 248169896, 993554240, 4086635408, 16350541128, 63529835824, 254129143040, 1046249323840, 4184725760584, 16276030608712, 65054467548432, 267635134298624

Offset: 1

Antti Karttunen, Sep 14 2006

Question: to which Wolfram's class does this simple program belong, class 3 or class 4, or is such categorization at all applicable here?

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, Terms a(1)-a(256) drawn as binary strings, showing details better.
Antti Karttunen, The central parts of terms a(1) - a(20000) drawn in similar fashion. (Note the "foreshadowing ghost columns" present in the "central skyscraper" attractor that are not present at A122245. Compare also to illustrations in A376412)
Index entries for sequences related to binary expansion of n
Index entries for encodings of plane rooted trees
Index entries for sequences related to parenthesizing

Cf. A014486, A057548, A082358, A122237, A122241, A122243 (same sequence in binary).
Compare to similar Wolframesque plots given in A080070, A122229, A122232, A122235, A122239, A122245.
Cf. also A376402, A376412.

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

For n >= 1, a(1+n) = 2*a(n) XOR A376402(n), a(4+n) = 16*a(n) XOR A376412(n). - Antti Karttunen, Sep 23 2024

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

A082358 Permutation of natural numbers: composition of permutations A057163 & A082356.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 5, 4, 6, 22, 21, 18, 17, 20, 13, 12, 10, 9, 11, 19, 16, 15, 14, 64, 63, 59, 58, 62, 50, 49, 46, 45, 48, 61, 57, 55, 54, 36, 35, 32, 31, 34, 27, 26, 24, 23, 25, 33, 30, 29, 28, 60, 56, 47, 44, 53, 52, 43, 41, 40, 51, 42, 38, 37, 39, 196, 195, 190, 189, 194

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

This is the signature-permutation of Catalan automorphism which is derived from nonrecursive Catalan automorphism *A123496 with the recursion schema FORK (defined in A122201). - Antti Karttunen, Oct 11 2006

Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082357. Cf. also A082359-A082360.
See the Wolframesque plots of A122242 and A122245.
Row 65796 of table A122201.

a(n) = A057163(A082356(n))

cp's OEIS Frontend

A179762 Binary expansions of A122245 (A122246) concatenated together to a single binary sequence, so that from each term of A122245, the most significant bits come before the least significant bits.

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A122246 a(n) = A007088(A122245(n)).

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A179776 Quadrisection of the fourth central column of triangle A122245, a(n) = A179775((4*n)-2).

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A179775 The fourth central column of triangle A122245, i.e., A179762(4), A179762(11), A179762(20), A179762(31), ...

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A376405 Bitwise XOR (centrally aligned) of two consecutive terms of A122245.

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PARI

Python

Formula

A376415 a(n) = A122245(4+n) XOR 16*A122245(n).

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PARI

Python

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A122242 a(n) = A014486(A122241(n)).

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

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A082358 Permutation of natural numbers: composition of permutations A057163 & A082356.

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