A122235 -id:A122235 - cp's OEIS frontend

A122236 a(n) = A007088(A122235(n)).

101100, 11011000, 1111001000, 111100010100, 11111100000100, 1111001001110000, 111100010111000100, 11111111000010000100, 1111100010011011000100, 111101101001110000100100

Offset: 1

Antti Karttunen, Sep 14 2006

Cf. A080070, A122230, A122233, A122240, A122243, A122246.

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

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

A122229 a(n) = A014486(A122228(n)).

Original entry on oeis.org

0, 2, 12, 56, 228, 920, 3684, 14744, 58980, 235928, 943716, 3774872, 15099492, 60397976, 241591908, 966367640, 3865470564, 15461882264, 61847529060, 247390116248, 989560464996, 3958241859992, 15832967439972

Offset: 0

Antti Karttunen, Sep 14 2006

A simple formula exists, cf. A080675.

A. Karttunen, Python program for computing this sequence and the associated image.
A. Karttunen, Terms a(1)-a(128) drawn as binary strings, in Wolframesque fashion.

A122230 shows the same sequence in binary. Compare to similar Wolframesque plots given in A080070, A122232, A122235, A122239, A122242, A122245.

A122232 a(n) = A014486(A122231(n)).

Original entry on oeis.org

42, 212, 992, 3876, 15448, 64644, 252056, 989988, 4108676, 16147220, 63393540, 266083460, 1047285272, 4245874244, 16903342544, 67034166420, 274274527940, 1068738181764, 4246566244100, 17369295361736, 67322784388376, 269731897678032

Offset: 1

Antti Karttunen, Sep 14 2006

A. Karttunen, Python program for computing this sequence and the associated image.
A. Karttunen, Terms a(1)-a(512) drawn as binary strings, in Wolframesque fashion.

A122233 shows the same sequence in binary. Compare to similar Wolframesque plots given in A080070, A122229, A122235, A122239, A122242, A122245.

A122239 a(n) = A014486(A122238(n)).

Original entry on oeis.org

52, 240, 964, 3972, 15556, 64532, 248288, 988964, 4164356, 15899248, 64719124, 257019652, 1070118936, 4197239188, 16299415152, 65592597568, 259741591312, 1093901323332, 4233842104068, 16616683414632, 70137217092164

Offset: 1

Antti Karttunen, Sep 14 2006

A122240 shows the same sequence in binary.

A. Karttunen, Python program for computing this sequence and the associated image.
A. Karttunen, Terms a(1)-a(512) drawn as binary strings, in Wolframesque fashion.

Compare to similar Wolframesque plots given in A080070, A122229, A122232, A122235, A122242, A122245.

A179417 a(n) is the binary number (shown here in decimal) constructed from quadratic residues of 65537 in range [(n^2)+1,(n+1)^2] in such a way that quadratic residues are mapped to 1-bits, and non-quadratic residues (as well as the multiples of 65537) to 0-bits, with the lower end of range mapped to less significant, and the higher end of range to more significant bits.

Original entry on oeis.org

1, 5, 24, 104, 279, 2001, 4131, 17453, 88826, 362532, 1655660, 6120642, 25376649, 128526482, 301370205, 1756488602, 8046359747, 30854867177, 73845140753, 488906501177, 2106640948770, 6573967883049, 29711211505300

Offset: 0

Antti Karttunen, Jul 27 2010

The binary width of terms are 1, 3, 5, 7, 9, ... i.e., the successive odd numbers, as their partial sums give the squares, 1, 4, 9, 16, ... at which points there certainly is always a quadratic residue, which thus gives the most significant bit for each number.

			In the range [(2^2)+1, (2+1)^2] (i.e., [5,9]) we have A165471(5)=A165471(6)=A165471(7)=-1 and A165471(8)=A165471(9)=+1, i.e., there are quadratic non-residues at points 5, 6 and 7, and quadratic residues at 8 and 9, so we construct a binary number 11000, which is 24 in decimal, thus a(2)=24.

Antti Karttunen, Table of n, a(n) for n = 0..256
Antti Karttunen, Terms a(0)-a(255) drawn as a bit triangle, 1 pixel per bit.
Antti Karttunen, Terms a(0)-a(255) drawn as a bit triangle, 2x2 pixels per bit.
Antti Karttunen, Terms a(0)-a(255) drawn as a bit triangle, 3x3 pixels per bit.

Cf. A179418.

Compare to similar bit triangle illustrations given in A080070, A122229, A122232, A122235, A122239, A122242, A122245.

A122234 Iterates of A122227, starting from A122227(5)=18.

Original entry on oeis.org

5, 18, 62, 180, 620, 1836, 5997, 23675, 76849, 263613, 923897, 3090855

Offset: 1

Antti Karttunen, Sep 14 2006

nonn

A122235.

cp's OEIS Frontend

A122236 a(n) = A007088(A122235(n)).

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

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

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

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A122229 a(n) = A014486(A122228(n)).

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A122232 a(n) = A014486(A122231(n)).

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A122239 a(n) = A014486(A122238(n)).

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A122234 Iterates of A122227, starting from A122227(5)=18.

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