A351003 Number of integer partitions y of n such that y_i = y_{i+1} for all even i.

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 28, 36, 42, 51, 62, 75, 88, 106, 124, 147, 173, 202, 236, 278, 320, 371, 431, 497, 572, 661, 756, 867, 993, 1132, 1291, 1474, 1672, 1898, 2155, 2439, 2756, 3117, 3512, 3957, 4458, 5008, 5624, 6316, 7072, 7919, 8862, 9899

Offset

0,3

Links

Table of n, a(n) for n=0..52.

Example

The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (311)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (411)     (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], And@@Table[#[[i]]==#[[i+1]], {i, 2, Length[#]-1, 2}]&]], {n, 0, 10}]

Crossrefs

The ordered version (compositions) is A027383.

The version for unequal instead of equal is A122135, even-length A351008.

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

Requiring inequalities at odd positions gives A351006, even-length A351007.

The even-length case is A351012.

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

Sequence in context: A350897 A008773 A008772 * A008771 A309831 A309830

Adjacent sequences: A351000 A351001 A351002 * A351004 A351005 A351006

Keyword

nonn

Author

Gus Wiseman, Jan 31 2022

Status

approved