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The version for even instead of odd positions indices is A351003, even-length A351012.

The version for even instead of odd positions is A351003, even-length A351012.

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Partitions These are partitions of n with all even multiplicities (or run-lengths), except possibly the last.

The version for alternately distinct unequal instead of equal parts is A122129, even-length A351008.

The opposite version for alternately distinct parts is A122135, even-length A122134 instead of odd positions is A351003.

The even instead of odd version is A351003.

Cf. A000070, A018819, A027383, 1A088218, A088218, A067661, A101417, `A305148, ~A339846, `A350837, `A350839, A122134, A122135, A350842, `A350844, ~A350948, A351007.

Also the number of integer partitions Partitions of n with all even multiplicities (or run-lengths) , except possibly the last, first A096441, both A349060.

Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

allocated Number of integer partitions y of n such that y_i = y_{i+1} for Gus Wisemanall odd i.

1, 1, 2, 2, 3, 3, 5, 4, 7, 7, 10, 9, 15, 13, 21, 19, 28, 26, 40, 35, 54, 49, 72, 64, 97, 87, 128, 115, 167, 151, 220, 195, 284, 256, 366, 328, 469, 421, 598, 537, 757, 682, 959, 859, 1204, 1085, 1507, 1354, 1880, 1694, 2338, 2104, 2892, 2609, 3574, 3218, 4394

0,3

Also the number of integer partitions of n with all even multiplicities (or run-lengths) except possibly the last, first A096441, both A349060.

The a(1) = 1 through a(9) = 7 partitions:

1 2 3 4 5 6 7 8 9

11 111 22 221 33 331 44 333

1111 11111 222 22111 332 441

2211 1111111 2222 22221

111111 3311 33111

221111 2211111

11111111 111111111

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

The ordered version (compositions) is A016116.

The even-length case is A035363.

A reverse version is A096441, both A349060.

The version for alternately distinct instead of equal parts is A122129, even-length A351008.

The opposite version for alternately distinct parts is A122135, even-length A122134.

The even instead of odd version is A351003.

The strict version is A351005, opposite A351006, even-length A035457.

Cf. A000070, A018819, A027383, 1A088218, A067661, A101417, `A305148, ~A339846, `A350837, `A350839, A350842, `A350844, ~A350948, A351007.

allocated

nonn

Gus Wiseman, Jan 31 2022

approved

editing

allocated for Gus Wiseman

allocated

approved