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 A351012 #5 Feb 10 2022 20:18:48
%S A351012 1,0,1,1,3,3,5,6,9,10,13,16,21,24,29,35,43,50,60,70,83,97,113,132,156,
%T A351012 178,206,239,275,316,365,416,477,545,620,706,806,912,1034,1173,1326,
%U A351012 1496,1691,1902,2141,2410,2704,3034,3406,3808,4261,4765,5317,5932,6617
%N A351012 Number of even-length integer partitions y of n such that y_i = y_{i+1} for all even i.
%e A351012 The a(2) = 1 through a(8) = 9 partitions:
%e A351012 (11) (21) (22) (32) (33) (43) (44)
%e A351012 (31) (41) (42) (52) (53)
%e A351012 (1111) (2111) (51) (61) (62)
%e A351012 (3111) (2221) (71)
%e A351012 (111111) (4111) (2222)
%e A351012 (211111) (3221)
%e A351012 (5111)
%e A351012 (311111)
%e A351012 (11111111)
%t A351012 Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,30}]
%Y A351012 The ordered version (compositions) is A027383(n-2).
%Y A351012 For odd instead of even indices we have A035363, any length A351004.
%Y A351012 The version for unequal parts appears to be A122134, any length A122135.
%Y A351012 This is the even-length case of A351003.
%Y A351012 Requiring inequalities at odd positions gives A351007, any length A351006.
%Y A351012 Cf. A000070, A018819, A035457, A088218, A101417, A122129, A350837, A351005, A351008.
%K A351012 nonn
%O A351012 0,5
%A A351012 _Gus Wiseman_, Feb 03 2022