A080070 -id:A080070 - cp's OEIS frontend

A080968 Orbit size of each branch-reduced tree encoded by A080981(n) under Donaghey's "Map M" Catalan automorphism.

1, 1, 2, 2, 3, 3, 3, 6, 6, 6, 6, 6, 6, 3, 2, 3, 5, 3, 5, 5, 5, 5, 3, 3, 2, 3, 6, 24, 24, 24, 24, 6, 24, 24, 24, 6, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 6, 24, 24, 24, 24, 24, 6, 24, 24, 6, 3, 18, 9, 24, 18, 18, 9, 18, 9, 18, 18, 3, 24, 15, 15, 24, 24, 18, 15, 15, 24, 3, 24, 24, 15, 15, 24

Offset: 0

Antti Karttunen, Mar 02 2003

nonn

This is the size of the cycle containing A080980(n) in the permutations A057505/A057506.

If the conjecture given in A080070 is true, then this sequence contains only six 2's. Questions: are there any (other) values with finite number of occurrences? Which primes will eventually appear?

Cf. A080969, A080972.

a(n) = A080967(A080980(n))

A126312 Fixed points of permutation A071661/A071662.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 22, 23, 30, 55, 64, 65, 98, 158, 196, 197, 318, 484, 625, 626, 687, 1042, 1549, 1973, 2055, 2056, 2376, 3471, 5113, 6558, 6917, 6918, 8191, 11763, 17268, 22277, 23713, 23714, 24331, 28360, 40491, 59362, 76942, 81754, 82499

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

Those i for which A071661(i)=i, i.e. for which A057163(A057164(i)) = A057164(A057163(i)). These appear to consist of just those general plane trees which are symmetric and will stay symmetric also after the underlying plane binary tree has been reflected, i.e. for which A057164(i)=i and A057164(A057163(i)) = A057163(i). See comments at A123050 and A080070. The sequence seems to give also the fixed points of the permutation A125977/A125978.

D. Callan, A Bijection on Dyck Paths and Its Cycle Structure, 2006, 17pp.

A154473 a(n) = A014486(A154472(n)).

Original entry on oeis.org

842, 11090, 13202, 46882, 60994, 231272004, 198873570440, 266349291297936, 64442911458703648, 3667589230123774528, 3336154829743802737792, 17601566387699271821281536, 1023499990310357893964861952

Offset: 0

Antti Karttunen, with terms a(0)-a(100) also independently computed by Wouter Meeussen, with the given Mathematica program, Jan 11 2009

This sequence gives the parenthesis expressions shown at the upper right corner image of the page 103 of NKS, with the left brackets (black squares) converted to 1's and the right brackets (white squares) converted to 0's and then interpreting each such number as a binary number and converted to decimal. A154474 shows the corresponding binary representations. Compare to A080070, A122242, A122245.

A. Karttunen and W. Meeussen, Table of n, a(n) for n = 0..100
S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 102, p. 103 and pages 104, 896-898.

Cf. A154475, A154476.

init=e[e[e][e]][e][e]
toDeca[ w_ ]:=FromDigits[ ToExpression[ Characters[ ToString[ w ] ]/.{"e"->Sequence[], "["->"1","]"->"0"} ],2 ]
toDeca /@ NestList[ #/.e[x_ ][y_ ]->x[x[y]]&, init, 100]

A154477 a(n) = A153240(A080068(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 2, 0, 1, 7, 4, 9, 8, 6, 7, 5, -6, -10, 16, 15, 10, -11, -15, 16, 22, 19, 24, 23, 18, 14, 28, 25, 21, 23, 11, -7, -26, 35, 34, 29, -18, 39, 38, 9, -8, 38, 33, -31, -35, 42, 37, 31, 32, 51, 48, -46, 54, 51, 40, -43, 58, 55, 43, 61, 60, 58, 52, 65, 62, -2, 68

Offset: 0

Antti Karttunen, Jan 11 2009

sign

This sequence gives some indication of how well the terms of A080068 are balanced as general trees, which has some implications as to the correctness of A123050 (see comments at A080070).

A. Karttunen, Table of n, a(n) for n = 0..512

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.

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.

A154474 a(n) = A007088(A154473(n)).

Original entry on oeis.org

1101001010, 10101101010010, 11001110010010, 1011011100100010, 1110111001000010, 1101110010001110111001000100, 10111001001101110010011101110010001000

Offset: 0

Antti Karttunen, Jan 11 2009

This sequence gives the parenthesis expressions shown at the upper right corner image of the page 103 of NKS, with left brackets (black squares) converted to 1's and right brackets (white squares) converted to 0's. Compare to A080070, A122242, A122245.

S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 102, p. 103 and pages 104, 896-898.

cp's OEIS Frontend

A080968 Orbit size of each branch-reduced tree encoded by A080981(n) under Donaghey's "Map M" Catalan automorphism.

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A126312 Fixed points of permutation A071661/A071662.

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A154473 a(n) = A014486(A154472(n)).

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A154477 a(n) = A153240(A080068(n)).

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

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A154474 a(n) = A007088(A154473(n)).

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