%I #7 Feb 06 2022 23:10:40

%S 1,0,0,1,1,2,2,3,4,5,5,7,8,9,10,13,14,16,18,20,23,27,28,32,37,40,44,

%T 51,54,60,67,73,81,90,96,107,118,127,139,154,166,181,198,213,232,256,

%U 273,297,325,348,377,411,440,476,516,555,598,647,692,746,807

%N Number of even-length integer partitions of n into parts that are alternately unequal and equal.

%C These are partitions whose multiplicities begin with a 1, are followed by any number of 2's, and end with another 1.

%e The a(3) = 1 through a(15) = 13 partitions (A..E = 10..14):

%e 21 31 32 42 43 53 54 64 65 75 76 86 87

%e 41 51 52 62 63 73 74 84 85 95 96

%e 61 71 72 82 83 93 94 A4 A5

%e 3221 81 91 92 A2 A3 B3 B4

%e 4221 5221 A1 B1 B2 C2 C3

%e 4331 4332 C1 D1 D2

%e 6221 5331 5332 5441 E1

%e 7221 6331 6332 5442

%e 8221 7331 6441

%e 9221 7332

%e 8331

%e A221

%e 433221

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

%Y The alternately equal and unequal version is A035457, any length A351005.

%Y This is the even-length case of A351006, odd-length A053251.

%Y Without equalities we have A351008, any length A122129, opposite A122135.

%Y Without inequalities we have A351012, any length A351003, opposite A351004.

%Y Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A350842, A350844, A351011.

%K nonn

%O 0,6

%A _Gus Wiseman_, Jan 31 2022