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