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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.

%I A345916 #5 Jul 10 2021 03:05:33
%S A345916 0,3,5,9,10,13,15,17,18,23,25,29,33,34,36,39,41,43,45,46,49,50,53,55,
%T A345916 57,58,61,63,65,66,68,71,75,77,78,81,85,89,90,95,97,98,103,105,109,
%U A345916 113,114,119,121,125,129,130,132,135,136,139,141,142,145,147,149
%N A345916 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum <= 0.
%C A345916 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A345916 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 A345916 The sequence of terms together with the corresponding compositions begins:
%e A345916      0: ()
%e A345916      3: (1,1)
%e A345916      5: (2,1)
%e A345916      9: (3,1)
%e A345916     10: (2,2)
%e A345916     13: (1,2,1)
%e A345916     15: (1,1,1,1)
%e A345916     17: (4,1)
%e A345916     18: (3,2)
%e A345916     23: (2,1,1,1)
%e A345916     25: (1,3,1)
%e A345916     29: (1,1,2,1)
%e A345916     33: (5,1)
%e A345916     34: (4,2)
%e A345916     36: (3,3)
%t A345916 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345916 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A345916 Select[Range[0,100],sats[stc[#]]<=0&]
%Y A345916 The version for Heinz numbers of partitions is A000290.
%Y A345916 These compositions are counted by A058622.
%Y A345916 These are the positions of terms <= 0 in A344618.
%Y A345916 The opposite (k >= 0) version is A345914.
%Y A345916 The version for unreversed alternating sum is A345915.
%Y A345916 The strictly negative (k < 0) version is A345920.
%Y A345916 A011782 counts compositions.
%Y A345916 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345916 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345916 A236913 counts partitions of 2n with reverse-alternating sum <= 0.
%Y A345916 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345916 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A345916 A345197 counts compositions by sum, length, and alternating sum.
%Y A345916 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345916 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345916 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345916 - k = 1:  counted by A000984, ranked by A345909/A345911.
%Y A345916 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345916 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345916 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345916 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345916 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345916 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345916 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345916 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345916 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345916 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345916 Cf. A000070, A000346, A008549, A025047, A027187, A028260, A032443, A114121, A163493, A344607, A344610, A345908.
%K A345916 nonn
%O A345916 1,2
%A A345916 _Gus Wiseman_, Jul 08 2021