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A345925 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 2.

%I A345925 #5 Jul 12 2021 18:02:17
%S A345925 2,9,11,14,34,37,39,42,45,47,52,57,59,62,132,137,139,142,146,149,151,
%T A345925 154,157,159,164,169,171,174,178,181,183,186,189,191,200,209,211,214,
%U A345925 220,226,229,231,234,237,239,244,249,251,254,520,529,531,534,540,546
%N A345925 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 2.
%C A345925 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A345925 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 A345925 The initial terms and corresponding compositions:
%e A345925       2: (2)            137: (4,3,1)
%e A345925       9: (3,1)          139: (4,2,1,1)
%e A345925      11: (2,1,1)        142: (4,1,1,2)
%e A345925      14: (1,1,2)        146: (3,3,2)
%e A345925      34: (4,2)          149: (3,2,2,1)
%e A345925      37: (3,2,1)        151: (3,2,1,1,1)
%e A345925      39: (3,1,1,1)      154: (3,1,2,2)
%e A345925      42: (2,2,2)        157: (3,1,1,2,1)
%e A345925      45: (2,1,2,1)      159: (3,1,1,1,1,1)
%e A345925      47: (2,1,1,1,1)    164: (2,3,3)
%e A345925      52: (1,2,3)        169: (2,2,3,1)
%e A345925      57: (1,1,3,1)      171: (2,2,2,1,1)
%e A345925      59: (1,1,2,1,1)    174: (2,2,1,1,2)
%e A345925      62: (1,1,1,1,2)    178: (2,1,3,2)
%e A345925     132: (5,3)          181: (2,1,2,2,1)
%t A345925 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A345925 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A345925 Select[Range[0,100],ats[stc[#]]==2&]
%Y A345925 These compositions are counted by A088218.
%Y A345925 These are the positions of 2's in A124754.
%Y A345925 The case of partitions of 2n is A344741.
%Y A345925 The version for reverse-alternating sum is A345922.
%Y A345925 The opposite (negative 2) version is A345924.
%Y A345925 The version for Heinz numbers of partitions is A345960 (reverse: A345961).
%Y A345925 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345925 A011782 counts compositions.
%Y A345925 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A345925 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345925 A120452 counts partitions of 2n with reverse-alternating sum 2.
%Y A345925 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A345925 A345197 counts compositions by sum, length, and alternating sum.
%Y A345925 Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
%Y A345925 Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
%Y A345925 - k = 0:  counted by A088218, ranked by A344619/A344619.
%Y A345925 - k = 1:  counted by A000984, ranked by A345909/A345911.
%Y A345925 - k = -1: counted by A001791, ranked by A345910/A345912.
%Y A345925 - k = 2:  counted by A088218, ranked by A345925/A345922.
%Y A345925 - k = -2: counted by A002054, ranked by A345924/A345923.
%Y A345925 - k >= 0: counted by A116406, ranked by A345913/A345914.
%Y A345925 - k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
%Y A345925 - k > 0:  counted by A027306, ranked by A345917/A345918.
%Y A345925 - k < 0:  counted by A294175, ranked by A345919/A345920.
%Y A345925 - k != 0: counted by A058622, ranked by A345921/A345921.
%Y A345925 - k even: counted by A081294, ranked by A053754/A053754.
%Y A345925 - k odd:  counted by A000302, ranked by A053738/A053738.
%Y A345925 Cf. A000070, A000097, A025047, A114121, A163493, A238279, A239830, A344607, A344608, A344609, A344651, A344743.
%K A345925 nonn
%O A345925 1,1
%A A345925 _Gus Wiseman_, Jul 11 2021