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A345915 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum <= 0.

%I A345915 #5 Jul 10 2021 03:05:26
%S A345915 0,3,6,10,12,13,15,20,24,25,27,30,36,40,41,43,46,48,49,50,51,53,54,55,
%T A345915 58,60,61,63,72,80,81,83,86,92,96,97,98,99,101,102,103,106,108,109,
%U A345915 111,116,120,121,123,126,136,144,145,147,150,156,160,161,162,163
%N A345915 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum <= 0.
%C A345915 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A345915 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 A345915 The sequence of terms together with the corresponding compositions begins:
%e A345915      0: ()
%e A345915      3: (1,1)
%e A345915      6: (1,2)
%e A345915     10: (2,2)
%e A345915     12: (1,3)
%e A345915     13: (1,2,1)
%e A345915     15: (1,1,1,1)
%e A345915     20: (2,3)
%e A345915     24: (1,4)
%e A345915     25: (1,3,1)
%e A345915     27: (1,2,1,1)
%e A345915     30: (1,1,1,2)
%e A345915     36: (3,3)
%e A345915     40: (2,4)
%e A345915     41: (2,3,1)
%t A345915 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345915 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345915 Select[Range[0,100],ats[stc[#]]<=0&]
%Y A345915 The version for Heinz numbers of partitions is A028260 (counted by A027187).
%Y A345915 These compositions are counted by A058622.
%Y A345915 These are the positions of terms <= 0 in A124754.
%Y A345915 The reverse-alternating version is A345916.
%Y A345915 The opposite (k >= 0) version is A345917.
%Y A345915 The strictly negative (k < 0) version is A345919.
%Y A345915 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345915 A011782 counts compositions.
%Y A345915 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345915 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345915 A236913 counts partitions of 2n with reverse-alternating sum <= 0.
%Y A345915 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345915 A345197 counts compositions by sum, length, and alternating sum.
%Y A345915 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345915 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345915 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345915 - k = 1:  counted by A000984, ranked by A345909/A345911.
%Y A345915 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345915 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345915 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345915 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345915 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345915 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345915 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345915 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345915 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345915 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345915 Cf. A000070, A000097, A000346, A008549, A025047, A032443, A114121, A163493, A344607, A344609, A344610.
%K A345915 nonn
%O A345915 1,2
%A A345915 _Gus Wiseman_, Jul 08 2021