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A345917 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum > 0.

%I A345917 #13 Nov 22 2021 19:59:33
%S A345917 1,2,4,5,7,8,9,11,14,16,17,18,19,21,22,23,26,28,29,31,32,33,34,35,37,
%T A345917 38,39,42,44,45,47,52,56,57,59,62,64,65,66,67,68,69,70,71,73,74,75,76,
%U A345917 77,78,79,82,84,85,87,88,89,90,91,93,94,95,100,104,105,107
%N A345917 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum > 0.
%C A345917 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A345917 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 A345917 The initial terms and the corresponding compositions:
%e A345917      1: (1)
%e A345917      2: (2)
%e A345917      4: (3)
%e A345917      5: (2,1)
%e A345917      7: (1,1,1)
%e A345917      8: (4)
%e A345917      9: (3,1)
%e A345917     11: (2,1,1)
%e A345917     14: (1,1,2)
%e A345917     16: (5)
%e A345917     17: (4,1)
%e A345917     18: (3,2)
%e A345917     19: (3,1,1)
%e A345917     21: (2,2,1)
%e A345917     22: (2,1,2)
%t A345917 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345917 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345917 Select[Range[0,100],ats[stc[#]]>0&]
%Y A345917 The version for Heinz numbers of partitions is A026424.
%Y A345917 These compositions are counted by A027306.
%Y A345917 These are the positions of terms > 0 in A124754.
%Y A345917 The weak (k >= 0) version is A345913.
%Y A345917 The reverse-alternating version is A345918.
%Y A345917 The opposite (k < 0) version is A345919.
%Y A345917 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345917 A011782 counts compositions.
%Y A345917 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345917 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345917 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345917 A345197 counts compositions by sum, length, and alternating sum.
%Y A345917 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345917 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345917 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345917 - k = 1:  counted by A000984, ranked by A345909/A345911.
%Y A345917 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345917 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345917 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345917 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345917 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345917 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345917 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345917 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345917 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345917 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345917 Cf. A000070, A000346, A008549, A025047, A027187, A027193, A032443, A114121, A163493, A344609, A345908.
%K A345917 nonn
%O A345917 1,2
%A A345917 _Gus Wiseman_, Jul 08 2021