A351012 Number of even-length integer partitions y of n such that y_i = y_{i+1} for all even i.

1, 0, 1, 1, 3, 3, 5, 6, 9, 10, 13, 16, 21, 24, 29, 35, 43, 50, 60, 70, 83, 97, 113, 132, 156, 178, 206, 239, 275, 316, 365, 416, 477, 545, 620, 706, 806, 912, 1034, 1173, 1326, 1496, 1691, 1902, 2141, 2410, 2704, 3034, 3406, 3808, 4261, 4765, 5317, 5932, 6617

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Example

The a(2) = 1 through a(8) = 9 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (3111)    (2221)    (71)
                              (111111)  (4111)    (2222)
                                        (211111)  (3221)
                                                  (5111)
                                                  (311111)
                                                  (11111111)

Mathematica

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

Crossrefs

The ordered version (compositions) is A027383(n-2).

For odd instead of even indices we have A035363, any length A351004.

The version for unequal parts appears to be A122134, any length A122135.

This is the even-length case of A351003.

Requiring inequalities at odd positions gives A351007, any length A351006.

Cf. A000070, A018819, A035457, A088218, A101417, A122129, A350837, A351005, A351008.

Sequence in context: A060022 A187679 A048274 * A059892 A241090 A091607

Adjacent sequences: A351009 A351010 A351011 * A351013 A351014 A351015

Keyword

nonn

Author

Gus Wiseman, Feb 03 2022

Status

approved