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

Author

Gus Wiseman, Jan 31 2022

Keywords

• nonn

Examples

			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)
		

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.

Programs

Mathematica

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