cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 41-47 of 47 results.

A129605 Signature-permutation of a Catalan automorphism, row 3613 of A089840.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 11, 14, 16, 19, 10, 15, 12, 17, 18, 13, 20, 21, 22, 23, 24, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 25, 29, 38, 43, 52, 26, 40, 31, 45, 46, 32, 48, 49, 50, 27, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 66, 67, 68, 69, 79, 80
Offset: 0

Views

Author

Antti Karttunen, May 22 2007

Keywords

Comments

This involution effects the following transformation on the binary trees (labels A,B,C,D refer to arbitrary subtrees located on those nodes and () stands for a terminal node.)
.....C...D.........A...D
......\./...........\./
...B...X2........C...Y2......B..().......A..()
....\./...........\./.........\./.........\./
.A...X1....-->.B...Y1......A...X1..-->.B...Y1
..\./...........\./.........\./.........\./
...X0............Y0..........X0..........Y0
Note that automorphism *A072796 = SPINE(*A129605). See the definition given in A122203.

Links

Crossrefs

Inverse: A129606.

A129607 Signature-permutation of a Catalan automorphism: swap the left and right subtree of degree 2 general trees.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11, 16, 19, 14, 15, 12, 17, 18, 13, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 44, 47, 33, 53, 56, 60, 37, 38, 39, 43, 52, 42, 40, 31, 45, 46, 32, 48, 49, 50, 51, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71
Offset: 0

Views

Author

Antti Karttunen, May 22 2007

Keywords

Comments

Otherwise like automorphism *A072796, except that this involution exchanges the two leftmost subtrees of a general tree ONLY when the degree of the tree is two. Automorphism *A129608 = SPINE(*A129607) = ENIPS(*A129607). See the definitions given in A122203 and A122204.

Links

Crossrefs

Row 3608 of A089840.

A129610 Signature-permutation of a Catalan automorphism, row 65352 of A089840.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 22 2007

Keywords

Comments

Automorphism *A074680 = SPINE(*A129610). See the definition given in A122203.

Links

Crossrefs

Inverse: A129609.

A130373 Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the first element in place if the length (A057515(n)) is odd.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 11, 10, 16, 19, 14, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 24, 30, 33, 37, 29, 26, 44, 47, 27, 53, 56, 60, 28, 39, 38, 43, 52, 42, 40, 31, 45, 46, 32, 48, 49, 50, 51, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 67, 66, 72, 75, 79, 71
Offset: 0

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

This self-inverse automorphism permutes the top level of a list of even length (1 2 3 4 ... 2n-1 2n) as (2 1 4 3 ... 2n 2n-1), and when applied to a list of odd length (1 2 3 4 5 ... 2n 2n+1), permutes it as (1 3 2 5 4 ... 2n+1 2n).

Links

Crossrefs

Cf. A057508, A057164, A057164.
SPINE and ENIPS transform of *A130340 (transformations explained in A122203 and A122204).
The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A073193 and A073192.

Formula

a(n) = A057508(A130374(A057508(n))) = A057164(A130374(A057164(n))).

A131303 2*A122204(deleting the right border of zeros) - A000012.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 7, 3, 3, 1, 9, 15, 5, 3, 1, 11, 13, 7, 5, 3, 1, 13, 11, 11, 9, 5, 3, 115, 7, 9, 7, 9, 5, 3, 1, 17, 9, 13, 11, 11, 11, 5, 3, 1, 19, 43, 15, 13, 7, 7, 7, 5, 3, 1
Offset: 1

Views

Author

Gary W. Adamson, Jun 27 2007

Keywords

Examples

			First few rows of the triangle:
   1;
   3,  1;
   5,  5,  1;
   7,  3,  3,  1;
   9, 15,  5,  3,  1;
  11, 13,  7,  5,  3,  1;
  13, 11, 11,  9,  5,  3,  1;
  ...

Crossrefs

Cf. A122204, A000012, A122203, A131302.

A131304 2*A089840 - A000012.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 7, 3, 3, 1, 9, 13, 5, 3, 1, 11, 15, 7, 5, 3, 1, 13, 11, 11, 9, 5, 3, 1, 15, 7, 9, 7, 9, 5, 3, 1, 17, 9, 13, 11, 11, 11, 5, 3, 1, 19, 33, 15, 137, 9, 11, 5, 3, 1
Offset: 1

Views

Author

Gary W. Adamson, Jun 27 2007

Keywords

Examples

			First few rows of the triangle:
   1;
   3,  1;
   5,  5,  1;
   7,  3,  3,  1;
   9, 13,  5,  3,  1;
  11, 15,  7,  5,  3,  1;
  13, 11, 11,  9,  5,  3,  1;
  ...

Crossrefs

Cf. A089840, A131302, A131303, A122203, A122204.

Formula

2*A089840(deleting the right border of zeros) - A000012.

A131305 2*A122288 - A000012.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 7, 3, 3, 1, 9, 15, 5, 3, 1, 11, 13, 7, 5, 3, 1, 13, 11, 11, 9, 5, 3, 1, 15, 7, 9, 7, 9, 5, 3, 1, 17, 9, 13, 11, 11, 11, 5, 3, 1, 19, 43, 15, 13, 7, 9, 11, 5, 3, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 27 2007

Keywords

Examples

			First few rows of the triangle:
   1;
   3,  1;
   5,  5,  1;
   7,  3,  3,  1;
   9, 15,  5,  3,  1;
  11, 13,  7,  5,  3,  1;
  13, 11, 11,  9,  5,  3,  1;
  ...

Crossrefs

Cf. A122288, A122203, A131302, A131303, A131304.

Formula

2*A122288(deleting the right border of zeros) - A000012.
Previous Showing 41-47 of 47 results.

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