cp's OEIS Frontend

A351594 Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.

%I A351594 #6 Mar 13 2022 19:00:49
%S A351594 0,1,1,2,1,3,2,4,2,7,3,9,4,13,6,19,6,26,10,35,12,49,16,64,20,87,27,
%T A351594 115,32,151,44,195,53,256,69,328,84,421,108,537,130,682,167,859,202,
%U A351594 1085,252,1354,305,1694,380,2104,456,2609,564,3218,676,3968,826,4863
%N A351594 Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.
%C A351594 These are partitions with all even run-lengths except for the last, which is odd.
%e A351594 The a(1) = 1 through a(9) = 7 partitions:
%e A351594   (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)    (9)
%e A351594             (111)       (221)    (222)  (331)      (332)  (333)
%e A351594                         (11111)         (22111)           (441)
%e A351594                                         (1111111)         (22221)
%e A351594                                                           (33111)
%e A351594                                                           (2211111)
%e A351594                                                           (111111111)
%t A351594 Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]
%Y A351594 The ordered version (compositions) is A016116 shifted right once.
%Y A351594 All odd-length partitions are counted by A027193.
%Y A351594 The opposite version is A117409, even-length A351012, any length A351003.
%Y A351594 Replacing equal with unequal relations appears to give:
%Y A351594 - any length: A122129
%Y A351594 - odd length: A122130
%Y A351594 - even length: A351008
%Y A351594 - opposite any length: A122135
%Y A351594 - opposite odd length: A351595
%Y A351594 - opposite even length: A122134
%Y A351594 This is the odd-length case of A351004, even-length A035363.
%Y A351594 The case that is also strict at even indices is:
%Y A351594 - any length: A351005
%Y A351594 - odd length: A351593
%Y A351594 - even length: A035457
%Y A351594 - opposite any length: A351006
%Y A351594 - opposite odd length: A053251
%Y A351594 - opposite even length: A351007
%Y A351594 A reverse version is A096441; see also A349060.
%Y A351594 Cf. A000009, A000041, A000070, A000984, A003242, A027383, A053738, A236559, A236914, A350842.
%K A351594 nonn
%O A351594 0,4
%A A351594 _Gus Wiseman_, Feb 24 2022