A122242 -id:A122242 - cp's OEIS frontend

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

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))

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.

A122241 Iterates of A122237, starting from 4.

Original entry on oeis.org

4, 22, 54, 169, 516, 1841, 6076, 19256, 66140, 252691, 888179, 2900616

Offset: 1

Antti Karttunen, Sep 14 2006

nonn

It might be more natural to use offset 0 for the initial value a(0) = 4 since that's not the result of an actual iteration of the map A122237. However, some other sequences (A122242, A179845, A179841, ...?) depend on this and have b-files that would require to be changed. - M. F. Hasler, Jul 18 2025

Cf. A122237 = A057548 o A082358, A122242 = A014486 o A122241.

(define (A122241 n) (if (= 1 n) 4 (A122237 (A122241 (- n 1)))))

a(n+1) = A122237(a(n)) for n >= 1; a(1) = 4.

A179757 a(n) = A014486(A179756(n)).

Original entry on oeis.org

56, 228, 932, 3736, 15512, 62040, 242264, 969136, 3985840, 15943080, 62096808, 248388496, 1021909904, 4087635248, 15883066672, 63532285248, 261545377088, 1046181434696, 4065904153928, 16263616920064, 66960253254144

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

A. Karttunen, Table of n, a(n) for n = 1..1024
A. Karttunen, Python program for computing this sequence and the associated images.
A. Karttunen, Terms a(1)-a(768) drawn as binary strings, one pixel per bit.
A. Karttunen, Terms a(1)-a(700) drawn as binary strings, 3x3 pixels per bit.

A122242, A122245, A179755. Cf. also A179763, A179828, A179829.

A179772 Position of ones in A179771.

Original entry on oeis.org

1, 2, 7, 8, 9, 13, 15, 20, 21, 24, 26, 27, 30, 31, 36, 38, 39, 47, 52, 56, 61, 63, 64, 65, 68, 69, 71, 73, 76, 81, 83, 84, 88, 93, 95, 96, 97, 101, 103, 105, 106, 110, 112, 114, 120, 125, 130, 137, 142, 144, 145, 146, 150, 152, 157, 158, 160, 161, 165, 167, 168

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

This seems to give the positions where "L"'s occur in the central column of A122242 (please zoom into the illustration given here). Conjecture, from a(60)=167 onward, all integers >= 167 present.

A. Karttunen, Table of n, a(n) for n = 1..150
A. Karttunen, Terms a(1)-a(700) of A122242 drawn as binary strings, 3x3 pixels per bit.

Cf. A179773-A179774, and also A179777, A179832.

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]

A179755 a(n) = A014486(A179754(n)).

Original entry on oeis.org

50, 216, 868, 3492, 13976, 56472, 225880, 897624, 3590576, 14471600, 57886120, 229868968, 919477136, 3706264464, 14825053488, 58832739632, 235330977088, 948740144448, 3794960504136, 15061020431688, 60244082031104

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

A. Karttunen, Table of n, a(n) for n = 1..1024
A. Karttunen, Python program for computing this sequence and the associated images.
A. Karttunen, Terms a(1)-a(768) drawn as binary strings, one pixel per bit.
A. Karttunen, Terms a(1)-a(700) drawn as binary strings, 3x3 pixels per bit.

a(n+1) = A004758(A179757(n)). Cf. also A122242, A122245.

cp's OEIS Frontend

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

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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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A122241 Iterates of A122237, starting from 4.

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Formula

A179757 a(n) = A014486(A179756(n)).

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A179772 Position of ones in A179771.

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

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Mathematica

A179755 a(n) = A014486(A179754(n)).

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