cp's OEIS Frontend

A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

%I A351004 #10 Feb 06 2022 23:10:28
%S A351004 1,1,2,2,3,3,5,4,7,7,10,9,15,13,21,19,28,26,40,35,54,49,72,64,97,87,
%T A351004 128,115,167,151,220,195,284,256,366,328,469,421,598,537,757,682,959,
%U A351004 859,1204,1085,1507,1354,1880,1694,2338,2104,2892,2609,3574,3218,4394
%N A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.
%C A351004 These are partitions of n with all even multiplicities (or run-lengths), except possibly the last.
%e A351004 The a(1) = 1 through a(9) = 7 partitions:
%e A351004   1  2   3    4     5      6       7        8         9
%e A351004      11  111  22    221    33      331      44        333
%e A351004               1111  11111  222     22111    332       441
%e A351004                            2211    1111111  2222      22221
%e A351004                            111111           3311      33111
%e A351004                                             221111    2211111
%e A351004                                             11111111  111111111
%t A351004 Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]
%Y A351004 The ordered version (compositions) is A016116.
%Y A351004 The even-length case is A035363.
%Y A351004 A reverse version is A096441, both A349060.
%Y A351004 The version for unequal instead of equal is A122129, even-length A351008.
%Y A351004 The version for even instead of odd indices is A351003, even-length A351012.
%Y A351004 The strict version is A351005, opposite A351006, even-length A035457.
%Y A351004 Cf. A000070, A018819, A027383, A088218, A067661, A101417, A122134, A122135, A350842, A351007.
%K A351004 nonn
%O A351004 0,3
%A A351004 _Gus Wiseman_, Jan 31 2022