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 A345915 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum <= 0. 25
 0, 3, 6, 10, 12, 13, 15, 20, 24, 25, 27, 30, 36, 40, 41, 43, 46, 48, 49, 50, 51, 53, 54, 55, 58, 60, 61, 63, 72, 80, 81, 83, 86, 92, 96, 97, 98, 99, 101, 102, 103, 106, 108, 109, 111, 116, 120, 121, 123, 126, 136, 144, 145, 147, 150, 156, 160, 161, 162, 163 (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..60. EXAMPLE The sequence of terms together with the corresponding compositions begins: 0: () 3: (1,1) 6: (1,2) 10: (2,2) 12: (1,3) 13: (1,2,1) 15: (1,1,1,1) 20: (2,3) 24: (1,4) 25: (1,3,1) 27: (1,2,1,1) 30: (1,1,1,2) 36: (3,3) 40: (2,4) 41: (2,3,1) 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 A028260 (counted by A027187). These compositions are counted by A058622. These are the positions of terms <= 0 in A124754. The reverse-alternating version is A345916. The opposite (k >= 0) version is A345917. The strictly negative (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). A236913 counts partitions of 2n with reverse-alternating sum <= 0. 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, A000097, A000346, A008549, A025047, A032443, A114121, A163493, A344607, A344609, A344610. Sequence in context: A282876 A363775 A261662 * A050107 A120068 A015875 Adjacent sequences: A345912 A345913 A345914 * A345916 A345917 A345918 KEYWORD nonn AUTHOR Gus Wiseman, Jul 08 2021 STATUS approved

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Last modified July 13 18:16 EDT 2024. Contains 374285 sequences. (Running on oeis4.)