This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A351007 #7 Feb 06 2022 23:10:40
%S A351007 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 A351007 51,54,60,67,73,81,90,96,107,118,127,139,154,166,181,198,213,232,256,
%U A351007 273,297,325,348,377,411,440,476,516,555,598,647,692,746,807
%N A351007 Number of even-length integer partitions of n into parts that are alternately unequal and equal.
%C A351007 These are partitions whose multiplicities begin with a 1, are followed by any number of 2's, and end with another 1.
%e A351007 The a(3) = 1 through a(15) = 13 partitions (A..E = 10..14):
%e A351007 21 31 32 42 43 53 54 64 65 75 76 86 87
%e A351007 41 51 52 62 63 73 74 84 85 95 96
%e A351007 61 71 72 82 83 93 94 A4 A5
%e A351007 3221 81 91 92 A2 A3 B3 B4
%e A351007 4221 5221 A1 B1 B2 C2 C3
%e A351007 4331 4332 C1 D1 D2
%e A351007 6221 5331 5332 5441 E1
%e A351007 7221 6331 6332 5442
%e A351007 8221 7331 6441
%e A351007 9221 7332
%e A351007 8331
%e A351007 A221
%e A351007 433221
%t A351007 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 A351007 The alternately equal and unequal version is A035457, any length A351005.
%Y A351007 This is the even-length case of A351006, odd-length A053251.
%Y A351007 Without equalities we have A351008, any length A122129, opposite A122135.
%Y A351007 Without inequalities we have A351012, any length A351003, opposite A351004.
%Y A351007 Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A350842, A350844, A351011.
%K A351007 nonn
%O A351007 0,6
%A A351007 _Gus Wiseman_, Jan 31 2022