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Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.
3

%I #6 Mar 13 2022 19:00:49

%S 0,1,1,2,1,3,2,4,2,7,3,9,4,13,6,19,6,26,10,35,12,49,16,64,20,87,27,

%T 115,32,151,44,195,53,256,69,328,84,421,108,537,130,682,167,859,202,

%U 1085,252,1354,305,1694,380,2104,456,2609,564,3218,676,3968,826,4863

%N Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.

%C These are partitions with all even run-lengths except for the last, which is odd.

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

%e (1) (2) (3) (4) (5) (6) (7) (8) (9)

%e (111) (221) (222) (331) (332) (333)

%e (11111) (22111) (441)

%e (1111111) (22221)

%e (33111)

%e (2211111)

%e (111111111)

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

%Y The ordered version (compositions) is A016116 shifted right once.

%Y All odd-length partitions are counted by A027193.

%Y The opposite version is A117409, even-length A351012, any length A351003.

%Y Replacing equal with unequal relations appears to give:

%Y - any length: A122129

%Y - odd length: A122130

%Y - even length: A351008

%Y - opposite any length: A122135

%Y - opposite odd length: A351595

%Y - opposite even length: A122134

%Y This is the odd-length case of A351004, even-length A035363.

%Y The case that is also strict at even indices is:

%Y - any length: A351005

%Y - odd length: A351593

%Y - even length: A035457

%Y - opposite any length: A351006

%Y - opposite odd length: A053251

%Y - opposite even length: A351007

%Y A reverse version is A096441; see also A349060.

%Y Cf. A000009, A000041, A000070, A000984, A003242, A027383, A053738, A236559, A236914, A350842.

%K nonn

%O 0,4

%A _Gus Wiseman_, Feb 24 2022