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A345923 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -2.

%I A345923 #6 Jul 12 2021 18:02:05
%S A345923 9,34,39,45,49,57,132,139,142,149,154,159,161,169,178,183,189,194,199,
%T A345923 205,209,217,226,231,237,241,249,520,531,534,540,549,554,559,564,571,
%U A345923 574,577,585,594,599,605,612,619,622,629,634,639,642,647,653,657,665
%N A345923 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -2.
%C A345923 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A345923 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 A345923 The initial terms and the corresponding compositions:
%e A345923       9: (3,1)            183: (2,1,2,1,1,1)
%e A345923      34: (4,2)            189: (2,1,1,1,2,1)
%e A345923      39: (3,1,1,1)        194: (1,5,2)
%e A345923      45: (2,1,2,1)        199: (1,4,1,1,1)
%e A345923      49: (1,4,1)          205: (1,3,1,2,1)
%e A345923      57: (1,1,3,1)        209: (1,2,4,1)
%e A345923     132: (5,3)            217: (1,2,1,3,1)
%e A345923     139: (4,2,1,1)        226: (1,1,4,2)
%e A345923     142: (4,1,1,2)        231: (1,1,3,1,1,1)
%e A345923     149: (3,2,2,1)        237: (1,1,2,1,2,1)
%e A345923     154: (3,1,2,2)        241: (1,1,1,4,1)
%e A345923     159: (3,1,1,1,1,1)    249: (1,1,1,1,3,1)
%e A345923     161: (2,5,1)          520: (6,4)
%e A345923     169: (2,2,3,1)        531: (5,3,1,1)
%e A345923     178: (2,1,3,2)        534: (5,2,1,2)
%t A345923 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345923 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A345923 Select[Range[0,100],sats[stc[#]]==-2&]
%Y A345923 These compositions are counted by A088218.
%Y A345923 These are the positions of 2's in A344618.
%Y A345923 The case of partitions of 2n is A344741.
%Y A345923 The opposite (negative 2) version is A345923.
%Y A345923 The version for unreversed alternating sum is A345925.
%Y A345923 The version for Heinz numbers of partitions is A345961.
%Y A345923 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345923 A011782 counts compositions.
%Y A345923 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345923 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345923 A120452 counts partitions of 2n with reverse-alternating sum 2.
%Y A345923 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345923 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A345923 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A345923 A345197 counts compositions by sum, length, and alternating sum.
%Y A345923 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345923 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345923 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345923 - k = 1:  counted by A000984, ranked by A345909/A345911.
%Y A345923 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345923 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345923 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345923 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345923 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345923 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345923 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345923 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345923 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345923 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345923 Cf. A000070, A000097, A000346, A025047, A032443, A034871, A106356, A163493, A236913, A344607, A344608, A344743.
%K A345923 nonn
%O A345923 1,1
%A A345923 _Gus Wiseman_, Jul 10 2021