A122229 -id:A122229 - cp's OEIS frontend

A122230 a(n) = A007088(A122229(n)).

0, 10, 1100, 111000, 11100100, 1110011000, 111001100100, 11100110011000, 1110011001100100, 111001100110011000, 11100110011001100100, 1110011001100110011000, 111001100110011001100100

Offset: 0

Antti Karttunen, Sep 14 2006

Cf. A080070, A122233, A122236, 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

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.

A122235 a(n) = A014486(A122234(n)).

Original entry on oeis.org

44, 216, 968, 3860, 16132, 62064, 247236, 1044612, 4073156, 16161828, 64513624, 253336008, 1046901060, 4267950372, 16347521428, 68075401492, 268150646664, 1086041921700, 4254535157576, 17346201751972, 66879000490408, 276319489325472

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.

A122236 shows the same sequence in binary. Compare to similar Wolframesque plots given in A080070, A122229, A122232, 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.

A080675 a(n) = (5*4^n - 8)/6.

Original entry on oeis.org

2, 12, 52, 212, 852, 3412, 13652, 54612, 218452, 873812, 3495252, 13981012, 55924052, 223696212, 894784852, 3579139412, 14316557652, 57266230612, 229064922452, 916259689812, 3665038759252, 14660155037012, 58640620148052, 234562480592212, 938249922368852

Offset: 1

N. J. A. Sloane, Mar 02 2003

These numbers have a simple binary pattern: 10,1100,110100,11010100,1101010100, ... i.e., the n-th term has a binary expansion 1(10){n-1}0, that is, there are n-1 10's between the most significant 1 and the least significant 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..170
Andrei Asinowski, Cyril Banderier, Benjamin Hackl, On extremal cases of pop-stack sorting, Permutation Patterns (Zürich, Switzerland, 2019).
Index entries for linear recurrences with constant coefficients, signature (5,-4).

a(n) = A072197(n-1) - 1 = A014486(|A106191(n)|). a(n) = A079946(A020988(n-2)) for n>=2. Cf. also A122229.

[(5*4^n-8)/6: n in [1..40] ]; // Vincenzo Librandi, Apr 28 2011

(5*4^Range[30]-8)/6 (* or *) LinearRecurrence[{5,-4},{2,12},30] (* Harvey P. Dale, Oct 16 2012 *)

a(n)=(5*4^n-8)/6 \\ Charles R Greathouse IV, Oct 07 2015

a(1)=2, a(2)=12, a(n)=5*a(n-1)-4*a(n-2). - Harvey P. Dale, Oct 16 2012

Further comments added by Antti Karttunen, Sep 14 2006

A122228 Iterates of A122227, starting from 0.

Original entry on oeis.org

0, 1, 3, 8, 20, 55, 160, 493, 1579, 5212, 17595, 60462, 210749, 743284, 2647461, 9509504

Offset: 0

Antti Karttunen, Sep 14 2006

nonn

Cf. A122229, A122230, A106191.

(define (A122228 n) (if (zero? n) 0 (A122227 (A122228 (- n 1)))))

cp's OEIS Frontend

A122230 a(n) = A007088(A122229(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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A122232 a(n) = A014486(A122231(n)).

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A122235 a(n) = A014486(A122234(n)).

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

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A080675 a(n) = (5*4^n - 8)/6.

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A122228 Iterates of A122227, starting from 0.

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