A351006 Number of integer partitions of n into parts that are alternately unequal and equal.

1, 1, 1, 2, 3, 4, 4, 6, 7, 9, 10, 12, 14, 16, 18, 22, 25, 28, 31, 36, 40, 46, 50, 56, 64, 71, 78, 88, 96, 106, 118, 130, 143, 158, 172, 190, 209, 228, 248, 274, 298, 324, 354, 384, 418, 458, 494, 536, 584, 631, 683, 742, 800, 864, 936, 1010, 1088, 1176, 1264

Offset: 0

Author

Gus Wiseman, Jan 31 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(11) = 12 partitions (A = 10, B = 11):
  1  2  3   4    5    6    7    8     9      A      B
        21  31   32   42   43   53    54     64     65
            211  41   51   52   62    63     73     74
                 311  411  61   71    72     82     83
                           322  422   81     91     92
                           511  611   522    433    A1
                                3221  711    622    533
                                      4221   811    722
                                      32211  5221   911
                                             42211  4331
                                                    6221
                                                    52211
		

Crossrefs

Without equalities we have A122129, opposite A122135, even-length A351008.

The non-strict version is A351003, opposite A351004, even-length A351012.

The alternately equal and unequal version is A351005, even-length A035457.

The even-length case is A351007.

Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A344605, A350842, A350844, A351011.

Programs

Mathematica

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