%I #6 Jul 12 2021 18:02:05

%S 9,34,39,45,49,57,132,139,142,149,154,159,161,169,178,183,189,194,199,

%T 205,209,217,226,231,237,241,249,520,531,534,540,549,554,559,564,571,

%U 574,577,585,594,599,605,612,619,622,629,634,639,642,647,653,657,665

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

%C The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

%C 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 The initial terms and the corresponding compositions:

%e 9: (3,1) 183: (2,1,2,1,1,1)

%e 34: (4,2) 189: (2,1,1,1,2,1)

%e 39: (3,1,1,1) 194: (1,5,2)

%e 45: (2,1,2,1) 199: (1,4,1,1,1)

%e 49: (1,4,1) 205: (1,3,1,2,1)

%e 57: (1,1,3,1) 209: (1,2,4,1)

%e 132: (5,3) 217: (1,2,1,3,1)

%e 139: (4,2,1,1) 226: (1,1,4,2)

%e 142: (4,1,1,2) 231: (1,1,3,1,1,1)

%e 149: (3,2,2,1) 237: (1,1,2,1,2,1)

%e 154: (3,1,2,2) 241: (1,1,1,4,1)

%e 159: (3,1,1,1,1,1) 249: (1,1,1,1,3,1)

%e 161: (2,5,1) 520: (6,4)

%e 169: (2,2,3,1) 531: (5,3,1,1)

%e 178: (2,1,3,2) 534: (5,2,1,2)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];

%t Select[Range[0,100],sats[stc[#]]==-2&]

%Y These compositions are counted by A088218.

%Y These are the positions of 2's in A344618.

%Y The case of partitions of 2n is A344741.

%Y The opposite (negative 2) version is A345923.

%Y The version for unreversed alternating sum is A345925.

%Y The version for Heinz numbers of partitions is A345961.

%Y A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

%Y A011782 counts compositions.

%Y A097805 counts compositions by alternating (or reverse-alternating) sum.

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A120452 counts partitions of 2n with reverse-alternating sum 2.

%Y A316524 gives the alternating sum of prime indices (reverse: A344616).

%Y A344610 counts partitions by sum and positive reverse-alternating sum.

%Y A344611 counts partitions of 2n with reverse-alternating sum >= 0.

%Y A345197 counts compositions by sum, length, and alternating sum.

%Y Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.

%Y Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:

%Y - k = 0: counted by A088218, ranked by A344619/A344619.

%Y - k = 1: counted by A000984, ranked by A345909/A345911.

%Y - k = -1: counted by A001791, ranked by A345910/A345912.

%Y - k = 2: counted by A088218, ranked by A345925/A345922.

%Y - k = -2: counted by A002054, ranked by A345924/A345923.

%Y - k >= 0: counted by A116406, ranked by A345913/A345914.

%Y - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.

%Y - k > 0: counted by A027306, ranked by A345917/A345918.

%Y - k < 0: counted by A294175, ranked by A345919/A345920.

%Y - k != 0: counted by A058622, ranked by A345921/A345921.

%Y - k even: counted by A081294, ranked by A053754/A053754.

%Y - k odd: counted by A000302, ranked by A053738/A053738.

%Y Cf. A000070, A000097, A000346, A025047, A032443, A034871, A106356, A163493, A236913, A344607, A344608, A344743.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 10 2021