# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a351007
Showing 1-1 of 1

%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

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE