A351005 Number of integer partitions of n into parts that are alternately equal and unequal.

1, 1, 2, 1, 2, 2, 3, 2, 4, 2, 5, 4, 6, 4, 8, 5, 10, 6, 12, 8, 16, 9, 18, 12, 22, 14, 28, 16, 33, 20, 40, 24, 48, 28, 56, 34, 67, 40, 80, 46, 94, 56, 110, 64, 130, 75, 152, 88, 176, 102, 206, 118, 238, 138, 276, 159, 320, 182, 368, 210, 424, 242, 488, 276, 558

Offset: 0

Author

Gus Wiseman, Jan 31 2022

Keywords

• nonn

Comments

Also partitions whose multiplicities are all 2's, except possibly for the last, which may be 1.

Examples

			The a(1) = 1 through a(12) = 6 partitions (A..C = 10..12):
  1  2   3  4   5    6     7    8     9    A     B      C
     11     22  221  33    331  44    441  55    443    66
                     2211       332        442   551    552
                                3311       3322  33221  4422
                                           4411         5511
                                                        332211
		

Crossrefs

The even-length ordered version is A003242, ranked by A351010.

The even-length case is A035457.

Without equalities we have A122135, opposite A122129, even-length A122134.

The non-strict version is A351004, opposite A351003, even-length A035363.

The opposite version is A351006, even-length A351007.

Cf. A000070, A018819, A027383, A088218, A350842, A350844, A351008.

Programs

Mathematica

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