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

Author

Gus Wiseman, Feb 24 2022

Keywords

• nonn

Comments

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

Examples

			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)
		

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.

Programs

Mathematica

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