Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
reviewed
approved
proposed
reviewed
editing
proposed
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
1,0,3
proposed
editing
editing
proposed
Also partitions whose multiplicities are all 2's, except possibly for the last, which may be 1.
allocated for Gus WisemanNumber of integer partitions of n 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
1,3
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
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}]
The even-length ordered version is A003242, ranked by A351010.
The even-length case is A035457.
Not requiring the equalities gives 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, `A087897, `A088218, `A101417, `A344605, ~A345194, ~A350837, ~A350839, A350842, A350844, A351008.
allocated
nonn
Gus Wiseman, Jan 31 2022
approved
editing
allocated for Gus Wiseman
allocated
approved