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A345914 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum >= 0.

%I A345914 #7 Jul 10 2021 03:05:10
%S A345914 0,1,2,3,4,6,7,8,10,11,12,13,14,15,16,19,20,21,22,24,26,27,28,30,31,
%T A345914 32,35,36,37,38,40,41,42,43,44,46,47,48,50,51,52,53,54,55,56,58,59,60,
%U A345914 61,62,63,64,67,69,70,72,73,74,76,79,80,82,83,84,86,87,88
%N A345914 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum >= 0.
%C A345914 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A345914 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 A345914 The sequence of terms together with the corresponding compositions begins:
%e A345914      0: ()           19: (3,1,1)        40: (2,4)
%e A345914      1: (1)          20: (2,3)          41: (2,3,1)
%e A345914      2: (2)          21: (2,2,1)        42: (2,2,2)
%e A345914      3: (1,1)        22: (2,1,2)        43: (2,2,1,1)
%e A345914      4: (3)          24: (1,4)          44: (2,1,3)
%e A345914      6: (1,2)        26: (1,2,2)        46: (2,1,1,2)
%e A345914      7: (1,1,1)      27: (1,2,1,1)      47: (2,1,1,1,1)
%e A345914      8: (4)          28: (1,1,3)        48: (1,5)
%e A345914     10: (2,2)        30: (1,1,1,2)      50: (1,3,2)
%e A345914     11: (2,1,1)      31: (1,1,1,1,1)    51: (1,3,1,1)
%e A345914     12: (1,3)        32: (6)            52: (1,2,3)
%e A345914     13: (1,2,1)      35: (4,1,1)        53: (1,2,2,1)
%e A345914     14: (1,1,2)      36: (3,3)          54: (1,2,1,2)
%e A345914     15: (1,1,1,1)    37: (3,2,1)        55: (1,2,1,1,1)
%e A345914     16: (5)          38: (3,1,2)        56: (1,1,4)
%t A345914 stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345914 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A345914 Select[Range[0,100],sats[stc[#]]>=0&]
%Y A345914 The version for prime indices is A000027, counted by A000041.
%Y A345914 These compositions are counted by A116406.
%Y A345914 The case of non-Heinz numbers of partitions is A119899, counted by A344608.
%Y A345914 The version for Heinz numbers of partitions is A344609, counted by A344607.
%Y A345914 These are the positions of terms >= 0 in A344618.
%Y A345914 The version for unreversed alternating sum is A345913.
%Y A345914 The opposite (k <= 0) version is A345916.
%Y A345914 The strict (k > 0) case is A345918.
%Y A345914 The complement is A345920, counted by A294175.
%Y A345914 A011782 counts compositions.
%Y A345914 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345914 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345914 A236913 counts partitions of 2n with reverse-alternating sum <= 0.
%Y A345914 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345914 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A345914 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A345914 A345197 counts compositions by sum, length, and alternating sum.
%Y A345914 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345914 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345914 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345914 - k = 1:  counted by A000984, ranked by A345909/A345911.
%Y A345914 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345914 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345914 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345914 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345914 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345914 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345914 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345914 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345914 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345914 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345914 Cf. A000070, A000346, A008549, A025047, A027187, A032443, A034871, A114121, A120452, A163493, A238279, A344650, A344743.
%K A345914 nonn
%O A345914 1,3
%A A345914 _Gus Wiseman_, Jul 04 2021