A080069 -id:A080069 - cp's OEIS frontend

A179758 Binary expansions of A080069 (A080070) concatenated together to a single binary sequence, so that from each term of A080069, the most significant bits come before the least significant bits.

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

Offset: 1

Antti Karttunen, Aug 03 2010

When viewed as a table, the first row contains two terms, the second four, the third six, and so on, i.e. row n contains 2n terms.

A. Karttunen, Table of n, a(n) for n = 1..262656 (containing 512 first rows)

Cf. also A179759 & A179761-A179763.

A328111 a(n) = A080069(n) OR A267357(n).

Original entry on oeis.org

1, 3, 15, 47, 191, 743, 2935, 12015, 47615, 190363, 737255, 3092431, 11777535, 48562151, 194672615, 778681963, 3117668351, 12677730147, 49850341191, 192901085003, 795560607711, 3243899871031, 12977889600367, 51055599708139, 204124618746111, 791262494980483, 3318011560984519, 12661179187462123, 52138250822737375, 212591566440951715, 836346216751952367, 3236342451194541807

Offset: 0

Antti Karttunen, Oct 05 2019

nonn

The pattern has a remarkably nice texture. A269174 gives the trajectory of 1-D Cellular Automaton rule 124 (which is a mirror image of rule 110), when started from a single alive cell. Trails of its evolution can be dimly discerned on the right hand side of given illustrations, while the left hand side shows the evolution of (left hand side of) iterated Dyck-path system A080069 unblemished.

Antti Karttunen, Table of n, a(n) for n = 0..1023
Antti Karttunen, Terms up to a(255) drawn as binary strings, with 1 bit = 3x3 pixels resolution
Antti Karttunen, Terms up to a(1023) drawn as binary strings, with 1 bit = 1 pixel resolution

Cf. A003986, A267357, A269174.
Cf. A080069, A080070, and also A079438 and A123050.
Cf. also A328103.

a(n) = A080069(n) OR A267357(n), where OR is bitwise-OR, A003986.

A057506 Signature-permutation of a Catalan Automorphism: (inverse of) "Donaghey's map M", acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 5, 4, 22, 19, 20, 15, 14, 21, 16, 18, 13, 11, 17, 12, 10, 9, 64, 60, 61, 52, 51, 62, 53, 55, 41, 39, 54, 40, 38, 37, 63, 56, 57, 43, 42, 59, 47, 50, 36, 33, 48, 34, 29, 28, 58, 44, 49, 35, 30, 46, 32, 27, 25, 45, 31, 26, 24, 23, 196, 191, 192, 178, 177

Offset: 0

Antti Karttunen, Sep 03 2000

This is inverse of A057505, which is a signature permutation of Catalan automorphism (bijection) known as "Donaghey's map M". See A057505 for more comments, links and references.

Antti Karttunen, Table of n, a(n) for n = 0..23713
A. Karttunen (et al) at OEIS Wiki, Maple implementations of CatalanRankGlobal, CatalanSequences, car, cdr, binexp2pars and pars2binexp functions
A. Karttunen at OEIS WIki, Scheme implementations of CatalanRankSexp and CatalanUnrankSexp
Indranil Ghosh, Python program for computing this sequence (after the functions mentioned in the OEIS wiki)
Index entries for signature-permutations of Catalan automorphisms

Inverse: A057505.
Cf. A057161, A057162, A057163, A057164, A057501, A057502, A057503, A057504 (for similar signature permutations of simple Catalan automorphisms).
Cf. A057507 (cycle counts).
The 2nd, 3rd, 4th, 5th and 6th "powers" of this permutation: A071662, A071664, A071666, A071668, A071670.
Row 12 of table A122287.
Cf. also A014486, A080300, A080069, A080070.

map(CatalanRankGlobal,map(DonagheysA057506,CatalanSequences(196))); # Where CatalanSequences(n) gives the terms A014486(0..n).
DonagheysA057506 := n -> pars2binexp(deepreverse(DonagheysA057505(deepreverse(binexp2pars(n)))));
DonagheysA057505 := h -> `if`((0 = nops(h)), h, [op(DonagheysA057505(car(h))), DonagheysA057505(cdr(h))]);
# The following corresponds to automorphism A057164:
deepreverse := proc(a) if 0 = nops(a) or list <> whattype(a) then (a) else [op(deepreverse(cdr(a))), deepreverse(a[1])]; fi; end;
# The rest of required Maple-functions: see the given OEIS Wiki page.

(define (A057506 n) (CatalanRankSexp (*A057506 (CatalanUnrankSexp n))))
(define (*A057506 bt) (let loop ((lt bt) (nt (list))) (cond ((not (pair? lt)) nt) (else (loop (cdr lt) (cons nt (*A057506 (car lt))))))))
;; Functions CatalanRankSexp and CatalanUnrankSexp can be found at OEIS Wiki page.

a(n) = A057163(A057164(n)).

a(n) = A057163(A057505(A057163(n))) = A057164(A057505(A057164(n))).

Entry revised by Antti Karttunen, May 30 2017

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.

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.

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.

cp's OEIS Frontend

A179758 Binary expansions of A080069 (A080070) concatenated together to a single binary sequence, so that from each term of A080069, the most significant bits come before the least significant bits.

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A328111 a(n) = A080069(n) OR A267357(n).

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A057506 Signature-permutation of a Catalan Automorphism: (inverse of) "Donaghey's map M", acting on the parenthesizations encoded by A014486.

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

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A122245 a(n) = A014486(A122244(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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A122235 a(n) = A014486(A122234(n)).

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

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