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%I #12 Jul 10 2021 03:04:57
%S 1,6,7,20,21,26,27,30,31,72,73,82,83,86,87,92,93,100,101,106,107,110,
%T 111,116,117,122,123,126,127,272,273,290,291,294,295,300,301,312,313,
%U 324,325,330,331,334,335,340,341,346,347,350,351,360,361,370,371,374
%N Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum 1.
%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 sequence of terms together with the corresponding compositions begins:
%e 1: (1)
%e 6: (1,2)
%e 7: (1,1,1)
%e 20: (2,3)
%e 21: (2,2,1)
%e 26: (1,2,2)
%e 27: (1,2,1,1)
%e 30: (1,1,1,2)
%e 31: (1,1,1,1,1)
%e 72: (3,4)
%e 73: (3,3,1)
%e 82: (2,3,2)
%e 83: (2,3,1,1)
%e 86: (2,2,1,2)
%e 87: (2,2,1,1,1)
%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[#]]==1&]
%Y These compositions are counted by A000984 (bisection of A126869).
%Y The version for Heinz numbers of partitions is A001105.
%Y A version using runs of binary digits is A066879.
%Y These are positions of 1's in A344618.
%Y The non-reverse version is A345909.
%Y The opposite (negative 1) version is A345912.
%Y The version for prime indices is A345958.
%Y Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%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 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 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, A027193, A031448, A034871, A114121, A120452, A344607.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jul 01 2021
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