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A351004
Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.
14
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
OFFSET
0,3
COMMENTS
These are partitions of n with all even multiplicities (or run-lengths), except possibly the last.
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.
Sequence in context: A036825 A035574 A036819 * A114328 A097366 A139807
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 31 2022
STATUS
approved