A351004 - OEIS

A351004

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

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OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..56.

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

The ordered version (compositions) is A016116.

The even-length case is A035363.

A reverse version is A096441, both A349060.

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

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

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

Cf. A000070, A018819, A027383, A088218, A067661, A101417, A122134, A122135, A350842, A351007.

Sequence in context: A036825 A035574 A036819 * A114328 A097366 A139807

Adjacent sequences: A351001 A351002 A351003 * A351005 A351006 A351007

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 31 2022

STATUS

approved