A082358 -id:A082358 - cp's OEIS frontend

A122237 a(n) = A057548(A082358(n)).

1, 3, 8, 7, 22, 21, 18, 17, 20, 64, 63, 59, 58, 62, 50, 49, 46, 45, 48, 61, 57, 55, 54, 196, 195, 190, 189, 194, 176, 175, 171, 170, 174, 193, 188, 185, 184, 148, 147, 143, 142, 146, 134, 133, 130, 129, 132, 145, 141, 139, 138, 192, 187, 173, 169, 183, 181, 167

Offset: 0

Antti Karttunen, Sep 14 2006

nonn

Iterates: A106191, A122241, A122244. Cf. also A122227, A080067.

A122201 Signature permutations of FORK-transformations of non-recursive Catalan automorphisms in table A089840.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 8, 3, 2, 1, 0, 6, 7, 4, 3, 2, 1, 0, 7, 6, 6, 5, 3, 2, 1, 0, 8, 5, 5, 4, 5, 3, 2, 1, 0, 9, 4, 7, 6, 6, 6, 3, 2, 1, 0, 10, 22, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 21, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 11, 12, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 18, 14, 13, 12

Offset: 0

Antti Karttunen, Sep 01 2006

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "FORK". In this recursion scheme the given automorphism is first applied at the root of binary tree, before the algorithm recurses down to the both branches (new ones, possibly changed by the given automorphism). I.e. this corresponds to the pre-order (prefix) traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures FORK and !FORK can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122202.

A. Karttunen, paper in preparation, draft available by e-mail.

Index entries for signature-permutations of Catalan automorphisms

The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A057163, 2: A057511, 3: A122341, 4: A122343, 5: A122345, 6: A122347, 7: A122349, 8: A082325, 9: A082360, 10: A122291, 11: A122293, 12: A074681, 13: A122295, 14: A122297, 15: A122353, 16: A122355, 17: A074684, 18: A122357, 19: A122359, 20: A122361, 21: A122301. Other rows: row 4253: A082356, row 65796: A082358, row 79361: A123493.

See also tables A089840, A122200, A122202-A122204, A122283-A122284, A122285-A122288, A122289-A122290.

(define (FORK foo) (letrec ((bar (lambda (s) (let ((t (foo s))) (if (pair? t) (cons (bar (car t)) (bar (cdr t))) t))))) bar))
(define (!FORK foo!) (letrec ((bar! (lambda (s) (cond ((pair? s) (foo! s) (bar! (car s)) (bar! (cdr s)))) s))) bar!))

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

A082355 Permutation of natural numbers induced by Catalan Automorphism *A082355 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 6, 7, 9, 10, 13, 11, 12, 19, 20, 21, 14, 15, 22, 18, 16, 17, 23, 24, 27, 25, 26, 33, 34, 35, 28, 29, 36, 32, 30, 31, 51, 52, 55, 53, 54, 56, 57, 58, 37, 38, 59, 41, 39, 40, 60, 61, 64, 47, 48, 62, 49, 42, 43, 63, 50, 46, 44, 45, 65, 66, 69, 67, 68

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant B quaternary encoding" as explained in the sequence A085184.

Inverse of A082356. a(n) = A082357(A057163(n)). a(n) = A082363(A082853(n))+A082852(n). Cf. also A082351-A082352, A082357-A082358.

Differs from A057118 first time at n=42: a(42)=56, while A057118(42)=58.

A082356 Permutation of natural numbers induced by Catalan Automorphism *A082356 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 6, 9, 10, 12, 13, 11, 17, 18, 21, 22, 20, 14, 15, 16, 19, 23, 24, 26, 27, 25, 31, 32, 35, 36, 34, 28, 29, 30, 33, 45, 46, 49, 50, 48, 58, 59, 63, 64, 62, 54, 55, 57, 61, 37, 38, 40, 41, 39, 42, 43, 44, 47, 51, 52, 56, 60, 53, 65, 66, 68, 69, 67

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant B quaternary encoding" as explained in the sequence A085184.

Inverse of A082355. a(n) = A057163(A082358(n)). a(n) = A082364(A082853(n))+A082852(n). Cf. also A082351-A082352, A082357-A082358.

Differs from A057117 first time at n=56: a(56)=42, while A057117(56)=44.

A082357 Permutation of natural numbers: composition of permutations A082355 & A057163.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 5, 4, 17, 16, 18, 15, 14, 22, 21, 20, 12, 11, 19, 13, 10, 9, 45, 44, 46, 43, 42, 50, 49, 48, 40, 39, 47, 41, 38, 37, 63, 62, 64, 59, 58, 61, 57, 54, 31, 30, 53, 32, 29, 28, 60, 56, 55, 36, 35, 52, 34, 26, 25, 51, 33, 27, 24, 23, 129, 128, 130, 127, 126

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082358. Cf. also A082359-A082360.

a(n) = A082355(A057163(n))

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.

A082359 Permutation of natural numbers: composition of permutations A074683 & A057163.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 6, 7, 9, 10, 13, 11, 12, 21, 22, 19, 14, 15, 20, 18, 16, 17, 23, 24, 27, 25, 26, 35, 36, 33, 28, 29, 34, 32, 30, 31, 58, 59, 64, 62, 63, 56, 60, 51, 37, 38, 52, 41, 39, 40, 57, 61, 55, 49, 50, 53, 47, 42, 43, 54, 48, 46, 44, 45, 65, 66, 69, 67, 68

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082360. Occurs in A073200 as row 18764713496857. Cf. also A082357-A082358.

a(n) = A074683(A057163(n)).

A082360 Permutation of natural numbers: composition of permutations A057163 & A074684.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 6, 9, 10, 12, 13, 11, 17, 18, 21, 22, 20, 16, 19, 14, 15, 23, 24, 26, 27, 25, 31, 32, 35, 36, 34, 30, 33, 28, 29, 45, 46, 49, 50, 48, 58, 59, 63, 64, 62, 57, 61, 54, 55, 44, 47, 56, 60, 53, 42, 51, 37, 38, 43, 52, 40, 41, 39, 65, 66, 68, 69, 67

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082359. Occurs in A073200. Cf. also A082357-A082358.

a(n) = A057163(A074684(n)).

A123496 Signature permutation of a nonrecursive Catalan automorphism: row 65796 of table A089840.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 4, 5, 6, 17, 18, 20, 21, 22, 9, 10, 11, 12, 13, 16, 19, 14, 15, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 44, 47, 53, 56, 60, 42, 51, 37, 38, 43, 52, 39, 40, 41, 129, 130, 132, 133, 134

Offset: 0

Antti Karttunen, Oct 11 2006

nonn

Inverse: A123495. a(n) = A069770(A082352(n)). Row 65796 of A089840. Used to construct automorphism *A082358. Cf. A069770 and A074680.

cp's OEIS Frontend

A122237 a(n) = A057548(A082358(n)).

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A122201 Signature permutations of FORK-transformations of non-recursive Catalan automorphisms in table A089840.

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

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A082355 Permutation of natural numbers induced by Catalan Automorphism *A082355 acting on the parenthesizations encoded by A014486/A063171.

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A082356 Permutation of natural numbers induced by Catalan Automorphism *A082356 acting on the parenthesizations encoded by A014486/A063171.

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A082357 Permutation of natural numbers: composition of permutations A082355 & A057163.

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A122241 Iterates of A122237, starting from 4.

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A082359 Permutation of natural numbers: composition of permutations A074683 & A057163.

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A082360 Permutation of natural numbers: composition of permutations A057163 & A074684.

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A123496 Signature permutation of a nonrecursive Catalan automorphism: row 65796 of table A089840.

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