cp's OEIS Frontend

A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

%I A355321 #7 Jun 30 2022 08:40:55
%S A355321 0,5,6,17,18,20,24,43,45,46,53,54,58,65,66,68,72,80,96,139,141,142,
%T A355321 149,150,154,163,165,166,169,172,177,178,180,184,197,198,202,209,210,
%U A355321 212,216,226,232,257,258,260,264,272,288,320,343,347,349,350,363,365
%N A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.
%C A355321 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%e A355321 The terms together with their corresponding compositions begin:
%e A355321    0: ()
%e A355321    5: (2,1)
%e A355321    6: (1,2)
%e A355321   17: (4,1)
%e A355321   18: (3,2)
%e A355321   20: (2,3)
%e A355321   24: (1,4)
%e A355321   43: (2,2,1,1)
%e A355321   45: (2,1,2,1)
%e A355321   46: (2,1,1,2)
%e A355321   53: (1,2,2,1)
%e A355321   54: (1,2,1,2)
%e A355321   58: (1,1,2,2)
%e A355321   65: (6,1)
%e A355321   66: (5,2)
%e A355321   68: (4,3)
%e A355321   72: (3,4)
%e A355321   80: (2,5)
%e A355321   96: (1,6)
%t A355321 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A355321 Select[Range[0,100],Count[stc[#],_?EvenQ]==Count[stc[#],_?OddQ]&]
%Y A355321 A subset of A001969 (evil numbers), complement A000069.
%Y A355321 These compositions are counted by A098123, without multiplicity A242821.
%Y A355321 The version for partitions is A325698, counted by A045931.
%Y A355321 For partitions without multiplicity we have A325700, counted by A241638.
%Y A355321 A047993 counts balanced partitions, ranked by A106529.
%Y A355321 A108950/A108949 count partitions with more odd/even parts.
%Y A355321 A130780/A171966 count partitions with more or as many odd/even parts.
%Y A355321 Cf. A000712, A001405, A026424, A028260, A239241 (conjugate A352129), A240009, A242498, A277579 (ranked by A349157).
%K A355321 nonn
%O A355321 1,2
%A A355321 _Gus Wiseman_, Jun 28 2022