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 A345916 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum <= 0. 25
 0, 3, 5, 9, 10, 13, 15, 17, 18, 23, 25, 29, 33, 34, 36, 39, 41, 43, 45, 46, 49, 50, 53, 55, 57, 58, 61, 63, 65, 66, 68, 71, 75, 77, 78, 81, 85, 89, 90, 95, 97, 98, 103, 105, 109, 113, 114, 119, 121, 125, 129, 130, 132, 135, 136, 139, 141, 142, 145, 147, 149 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) 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 EXAMPLE The sequence of terms together with the corresponding compositions begins:      0: ()      3: (1,1)      5: (2,1)      9: (3,1)     10: (2,2)     13: (1,2,1)     15: (1,1,1,1)     17: (4,1)     18: (3,2)     23: (2,1,1,1)     25: (1,3,1)     29: (1,1,2,1)     33: (5,1)     34: (4,2)     36: (3,3) MATHEMATICA stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse; sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}]; Select[Range[0, 100], sats[stc[#]]<=0&] CROSSREFS The version for Heinz numbers of partitions is A000290. These compositions are counted by A058622. These are the positions of terms <= 0 in A344618. The opposite (k >= 0) version is A345914. The version for unreversed alternating sum is A345915. The strictly negative (k < 0) version is A345920. 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). A344611 counts partitions of 2n with reverse-alternating sum >= 0. 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, A028260, A032443, A114121, A163493, A344607, A344610, A345908. Sequence in context: A153710 A230385 A269399 * A175468 A286065 A316296 Adjacent sequences:  A345913 A345914 A345915 * A345917 A345918 A345919 KEYWORD nonn AUTHOR Gus Wiseman, Jul 08 2021 STATUS approved

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Last modified August 15 02:47 EDT 2022. Contains 356122 sequences. (Running on oeis4.)