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%I #8 Jul 12 2021 18:02:11
%S 12,40,49,51,54,60,144,161,163,166,172,184,194,197,199,202,205,207,
%T 212,217,219,222,232,241,243,246,252,544,577,579,582,588,600,624,642,
%U 645,647,650,653,655,660,665,667,670,680,689,691,694,700,720,737,739,742
%N Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -2.
%C The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) 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 12: (1,3) 202: (1,3,2,2) 582: (3,4,1,2)
%e 40: (2,4) 205: (1,3,1,2,1) 588: (3,3,1,3)
%e 49: (1,4,1) 207: (1,3,1,1,1,1) 600: (3,2,1,4)
%e 51: (1,3,1,1) 212: (1,2,2,3) 624: (3,1,1,5)
%e 54: (1,2,1,2) 217: (1,2,1,3,1) 642: (2,6,2)
%e 60: (1,1,1,3) 219: (1,2,1,2,1,1) 645: (2,5,2,1)
%e 144: (3,5) 222: (1,2,1,1,1,2) 647: (2,5,1,1,1)
%e 161: (2,5,1) 232: (1,1,2,4) 650: (2,4,2,2)
%e 163: (2,4,1,1) 241: (1,1,1,4,1) 653: (2,4,1,2,1)
%e 166: (2,3,1,2) 243: (1,1,1,3,1,1) 655: (2,4,1,1,1,1)
%e 172: (2,2,1,3) 246: (1,1,1,2,1,2) 660: (2,3,2,3)
%e 184: (2,1,1,4) 252: (1,1,1,1,1,3) 665: (2,3,1,3,1)
%e 194: (1,5,2) 544: (4,6) 667: (2,3,1,2,1,1)
%e 197: (1,4,2,1) 577: (3,6,1) 670: (2,3,1,1,1,2)
%e 199: (1,4,1,1,1) 579: (3,5,1,1) 680: (2,2,2,4)
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t Select[Range[0,100],ats[stc[#]]==-2&]
%Y These compositions are counted by A002054.
%Y These are the positions of -2's in A124754.
%Y The version for reverse-alternating sum is A345923.
%Y The opposite (positive 2) version is A345925.
%Y The version for Heinz numbers of partitions is A345962.
%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 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, A008549, A025047, A163493, A239830, A344609, A344651, A344741, A345908, A345961.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jul 11 2021
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