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A344619 The a(n)-th composition in standard order (A066099) has alternating sum 0.

%I A344619 #7 Jun 08 2021 07:26:47
%S A344619 0,3,10,13,15,36,41,43,46,50,53,55,58,61,63,136,145,147,150,156,162,
%T A344619 165,167,170,173,175,180,185,187,190,196,201,203,206,210,213,215,218,
%U A344619 221,223,228,233,235,238,242,245,247,250,253,255,528,545,547,550,556,568
%N A344619 The a(n)-th composition in standard order (A066099) has alternating sum 0.
%C A344619 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A344619 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 A344619 The sequence of terms together with the corresponding compositions begins:
%e A344619     0: ()
%e A344619     3: (1,1)
%e A344619    10: (2,2)
%e A344619    13: (1,2,1)
%e A344619    15: (1,1,1,1)
%e A344619    36: (3,3)
%e A344619    41: (2,3,1)
%e A344619    43: (2,2,1,1)
%e A344619    46: (2,1,1,2)
%e A344619    50: (1,3,2)
%e A344619    53: (1,2,2,1)
%e A344619    55: (1,2,1,1,1)
%e A344619    58: (1,1,2,2)
%e A344619    61: (1,1,1,2,1)
%e A344619    63: (1,1,1,1,1,1)
%e A344619   136: (4,4)
%e A344619   145: (3,4,1)
%e A344619   147: (3,3,1,1)
%e A344619   150: (3,2,1,2)
%e A344619   156: (3,1,1,3)
%t A344619 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A344619 stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]]
%t A344619 Select[Range[0,100],ats[stc[#]]==0&]
%Y A344619 The version for Heinz numbers of partitions is A000290, counted by A000041.
%Y A344619 These are the positions of zeros in A344618.
%Y A344619 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A344619 A116406 counts compositions with alternating sum >= 0.
%Y A344619 A124754 gives the alternating sum of standard compositions.
%Y A344619 A316524 is the alternating sum of the prime indices of n.
%Y A344619 A344604 counts wiggly compositions with twins.
%Y A344619 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A344619 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A344619 A344616 gives the alternating sum of reversed prime indices.
%Y A344619 All of the following pertain to compositions in standard order:
%Y A344619 - The length is A000120.
%Y A344619 - Converting to reversed ranking gives A059893.
%Y A344619 - The rows are A066099.
%Y A344619 - The sum is A070939.
%Y A344619 - The runs are counted by A124767.
%Y A344619 - The reversed version is A228351.
%Y A344619 - Strict compositions are ranked by A233564.
%Y A344619 - Constant compositions are ranked by A272919.
%Y A344619 - The Heinz number is A333219.
%Y A344619 - Anti-run compositions are ranked by A333489.
%Y A344619 Cf. A000070, A000097, A003242, A006330, A028260, A119899, A239830, A344605, A344607, A344650, A344739.
%K A344619 nonn
%O A344619 1,2
%A A344619 _Gus Wiseman_, Jun 03 2021