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 A345917 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum > 0. 31
 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, 38, 39, 42, 44, 45, 47, 52, 56, 57, 59, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 84, 85, 87, 88, 89, 90, 91, 93, 94, 95, 100, 104, 105, 107 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. 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. LINKS Table of n, a(n) for n=1..66. EXAMPLE The initial terms and the corresponding compositions: 1: (1) 2: (2) 4: (3) 5: (2,1) 7: (1,1,1) 8: (4) 9: (3,1) 11: (2,1,1) 14: (1,1,2) 16: (5) 17: (4,1) 18: (3,2) 19: (3,1,1) 21: (2,2,1) 22: (2,1,2) MATHEMATICA stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse; ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}]; Select[Range[0, 100], ats[stc[#]]>0&] CROSSREFS The version for Heinz numbers of partitions is A026424. These compositions are counted by A027306. These are the positions of terms > 0 in A124754. The weak (k >= 0) version is A345913. The reverse-alternating version is A345918. The opposite (k < 0) version is A345919. A000041 counts partitions of 2n with alternating sum 0, ranked by A000290. A011782 counts compositions. A097805 counts compositions by alternating (or reverse-alternating) sum. A103919 counts partitions by sum and alternating sum (reverse: A344612). A316524 gives the alternating sum of prime indices (reverse: A344616). A345197 counts compositions by sum, length, and alternating sum. Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618. Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k: - k = 0: counted by A088218, ranked by A344619/A344619. - k = 1: counted by A000984, ranked by A345909/A345911. - k = -1: counted by A001791, ranked by A345910/A345912. - k = 2: counted by A088218, ranked by A345925/A345922. - k = -2: counted by A002054, ranked by A345924/A345923. - k >= 0: counted by A116406, ranked by A345913/A345914. - k <= 0: counted by A058622(n-1), ranked by A345915/A345916. - k > 0: counted by A027306, ranked by A345917/A345918. - k < 0: counted by A294175, ranked by A345919/A345920. - k != 0: counted by A058622, ranked by A345921/A345921. - k even: counted by A081294, ranked by A053754/A053754. - k odd: counted by A000302, ranked by A053738/A053738. Cf. A000070, A000346, A008549, A025047, A027187, A027193, A032443, A114121, A163493, A344609, A345908. Sequence in context: A187351 A026451 A050106 * A303739 A183544 A219640 Adjacent sequences: A345914 A345915 A345916 * A345918 A345919 A345920 KEYWORD nonn AUTHOR Gus Wiseman, Jul 08 2021 STATUS approved

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Last modified June 14 05:27 EDT 2024. Contains 373393 sequences. (Running on oeis4.)