cp's OEIS Frontend

A345910 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -1.

%I A345910 #9 Jul 09 2021 22:57:44
%S A345910 6,20,25,27,30,72,81,83,86,92,98,101,103,106,109,111,116,121,123,126,
%T A345910 272,289,291,294,300,312,322,325,327,330,333,335,340,345,347,350,360,
%U A345910 369,371,374,380,388,393,395,398,402,405,407,410,413,415,420,425,427
%N A345910 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -1.
%C A345910 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A345910 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 A345910 The sequence of terms together with the corresponding compositions begins:
%e A345910       6: (1,2)
%e A345910      20: (2,3)
%e A345910      25: (1,3,1)
%e A345910      27: (1,2,1,1)
%e A345910      30: (1,1,1,2)
%e A345910      72: (3,4)
%e A345910      81: (2,4,1)
%e A345910      83: (2,3,1,1)
%e A345910      86: (2,2,1,2)
%e A345910      92: (2,1,1,3)
%e A345910      98: (1,4,2)
%e A345910     101: (1,3,2,1)
%e A345910     103: (1,3,1,1,1)
%e A345910     106: (1,2,2,2)
%e A345910     109: (1,2,1,2,1)
%t A345910 stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345910 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345910 Select[Range[0,100],ats[stc[#]]==-1&]
%Y A345910 These compositions are counted by A001791.
%Y A345910 A version using runs of binary digits is A031444.
%Y A345910 These are the positions of -1's in A124754.
%Y A345910 The opposite (positive 1) version is A345909.
%Y A345910 The reverse version is A345912.
%Y A345910 The version for alternating sum of prime indices is A345959.
%Y A345910 Standard compositions: A000120, A066099, A070939, A124754, A228351, A344618.
%Y A345910 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345910 A000070 counts partitions of 2n+1 with alternating sum 1, ranked by A001105.
%Y A345910 A011782 counts compositions.
%Y A345910 A097805 counts compositions by sum and alternating sum.
%Y A345910 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345910 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345910 A345197 counts compositions by sum, length, and alternating sum.
%Y A345910 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345910 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345910 - k = 1:  counted by A000984, ranked by A345909/A345911.
%Y A345910 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345910 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345910 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345910 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345910 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345910 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345910 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345910 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345910 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345910 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345910 Cf. A000097, A000346, A008549, A025047, A027187, A031443, A031448, A114121, A119899, A126869, A238279, A344617.
%K A345910 nonn
%O A345910 1,1
%A A345910 _Gus Wiseman_, Jul 01 2021