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Number of integer partitions y of n such that y_i = y_{i+1} for all even i.
18

%I #8 Feb 06 2022 23:10:19

%S 1,1,2,3,5,6,9,11,15,18,23,28,36,42,51,62,75,88,106,124,147,173,202,

%T 236,278,320,371,431,497,572,661,756,867,993,1132,1291,1474,1672,1898,

%U 2155,2439,2756,3117,3512,3957,4458,5008,5624,6316,7072,7919,8862,9899

%N Number of integer partitions y of n such that y_i = y_{i+1} for all even i.

%e The a(1) = 1 through a(7) = 11 partitions:

%e (1) (2) (3) (4) (5) (6) (7)

%e (11) (21) (22) (32) (33) (43)

%e (111) (31) (41) (42) (52)

%e (211) (311) (51) (61)

%e (1111) (2111) (222) (322)

%e (11111) (411) (511)

%e (3111) (2221)

%e (21111) (4111)

%e (111111) (31111)

%e (211111)

%e (1111111)

%t Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,10}]

%Y The ordered version (compositions) is A027383.

%Y The version for unequal instead of equal is A122135, even-length A351008.

%Y For odd instead of even indices we have A351004, even-length A035363.

%Y Requiring inequalities at odd positions gives A351006, even-length A351007.

%Y The even-length case is A351012.

%Y Cf. A000070, A018819, A088218, A101417, A122129, A350837, A351005.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 31 2022