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A351005 revision #8

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

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OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..64.

EXAMPLE

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

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}]

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.

Sequence in context: A283451 A372970 A172245 * A238781 A319439 A051275

Adjacent sequences: A351002 A351003 A351004 * A351006 A351007 A351008

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 31 2022

STATUS

reviewed