A351594 Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.

0, 1, 1, 2, 1, 3, 2, 4, 2, 7, 3, 9, 4, 13, 6, 19, 6, 26, 10, 35, 12, 49, 16, 64, 20, 87, 27, 115, 32, 151, 44, 195, 53, 256, 69, 328, 84, 421, 108, 537, 130, 682, 167, 859, 202, 1085, 252, 1354, 305, 1694, 380, 2104, 456, 2609, 564, 3218, 676, 3968, 826, 4863

Offset

0,4

Comments

These are partitions with all even run-lengths except for the last, which is odd.

Links

Table of n, a(n) for n=0..59.

Example

The a(1) = 1 through a(9) = 7 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)    (9)
            (111)       (221)    (222)  (331)      (332)  (333)
                        (11111)         (22111)           (441)
                                        (1111111)         (22221)
                                                          (33111)
                                                          (2211111)
                                                          (111111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]], {i, 1, Length[#]-1, 2}]&]], {n, 0, 30}]

Crossrefs

The ordered version (compositions) is A016116 shifted right once.

All odd-length partitions are counted by A027193.

The opposite version is A117409, even-length A351012, any length A351003.

Replacing equal with unequal relations appears to give:

- any length: A122129

- odd length: A122130

- even length: A351008

- opposite any length: A122135

- opposite odd length: A351595

- opposite even length: A122134

This is the odd-length case of A351004, even-length A035363.

The case that is also strict at even indices is:

- any length: A351005

- odd length: A351593

- even length: A035457

- opposite any length: A351006

- opposite odd length: A053251

- opposite even length: A351007

A reverse version is A096441; see also A349060.

Cf. A000009, A000041, A000070, A000984, A003242, A027383, A053738, A236559, A236914, A350842.

Sequence in context: A294199 A078658 A307719 * A185314 A285120 A282744

Adjacent sequences: A351591 A351592 A351593 * A351595 A351596 A351597

Keyword

nonn

Author

Gus Wiseman, Feb 24 2022

Status

approved