A317303 Numbers k such that both Dyck paths of the symmetric representation of sigma(k) have a central peak.

2, 7, 8, 9, 16, 17, 18, 19, 20, 29, 30, 31, 32, 33, 34, 35, 46, 47, 48, 49, 50, 51, 52, 53, 54, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 154, 155, 156, 157, 158, 159, 160

Offset

1,1

Comments

Also triangle read by rows which gives the odd-indexed rows of triangle A014132.

There are no triangular number (A000217) in this sequence.

For more information about the symmetric representation of sigma see A237593 and its related sequences.

Equivalently, numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have an odd number of peaks. - Omar E. Pol, Sep 13 2018

Links

Table of n, a(n) for n=1..71.

Example

Written as an irregular triangle in which the row lengths are the odd numbers, the sequence begins:
    2;
    7,   8,   9;
   16,  17,  18,  19,  20;
   29,  30,  31,  32,  33,  34,  35;
   46,  47,  48,  49,  50,  51,  52,  53,  54;
   67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77;
   92,  93,  94,  95,  96,  97,  98,  99, 100, 101, 102, 103, 104;
  121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135;
...
Illustration of initial terms:
-----------------------------------------------------------
   k   sigma(k)   Diagram of the symmetry of sigma
-----------------------------------------------------------
                    _         _ _ _             _ _ _ _ _
                  _| |       | | | |           | | | | | |
   2      3      |_ _|       | | | |           | | | | | |
                             | | | |           | | | | | |
                            _|_| | |           | | | | | |
                          _|  _ _|_|           | | | | | |
                  _ _ _ _|  _| |               | | | | | |
   7      8      |_ _ _ _| |_ _|               | | | | | |
   8     15      |_ _ _ _ _|              _ _ _| | | | | |
   9     13      |_ _ _ _ _|             |  _ _ _|_| | | |
                                        _| |    _ _ _|_| |
                                      _|  _|   |  _ _ _ _|
                                  _ _|  _|  _ _| |
                                 |  _ _|  _|    _|
                                 | |     |     |
                  _ _ _ _ _ _ _ _| |  _ _|  _ _|
  16     31      |_ _ _ _ _ _ _ _ _| |  _ _|
  17     18      |_ _ _ _ _ _ _ _ _| | |
  18     39      |_ _ _ _ _ _ _ _ _ _| |
  19     20      |_ _ _ _ _ _ _ _ _ _| |
  20     42      |_ _ _ _ _ _ _ _ _ _ _|
.
For the first nine terms of the sequence we can see in the above diagram that both Dyck path (the smallest and the largest) of the symmetric representation of sigma(k) have a central peak.
Compare with A317304.

Crossrefs

Column 1 gives A130883, n >= 1.

Column 2 gives A033816, n >= 1.

Row sums give the odd-indexed terms of A006002.

Right border gives the positive terms of A014107, also the odd-indexed terms of A000096.

The union of A000217, A317304 and this sequence gives A001477.

Some other sequences related to the central peak or the central valley of the symmetric representation of sigma are A000217, A000384, A007606, A007607, A014105, A014132, A162917, A161983, A317304. See also A317306.

Cf. A000203, A005408, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A249351, A262626.

Sequence in context: A047354 A037455 A020675 * A045980 A104339 A199004

Adjacent sequences: A317300 A317301 A317302 * A317304 A317305 A317306

Keyword

nonn,tabf

Author

Omar E. Pol, Aug 27 2018

Status

approved