%I #6 Jul 12 2021 18:01:52

%S 1,2,4,5,6,7,8,9,11,12,14,16,17,18,19,20,21,22,23,24,25,26,27,28,29,

%T 30,31,32,33,34,35,37,38,39,40,42,44,45,47,48,49,51,52,54,56,57,59,60,

%U 62,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81

%N Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum != 0.

%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.

%C Also numbers k such that the k-th composition in standard order has reverse-alternating sum != 0.

%e The initial terms and the corresponding compositions:

%e 1: (1) 20: (2,3) 35: (4,1,1)

%e 2: (2) 21: (2,2,1) 37: (3,2,1)

%e 4: (3) 22: (2,1,2) 38: (3,1,2)

%e 5: (2,1) 23: (2,1,1,1) 39: (3,1,1,1)

%e 6: (1,2) 24: (1,4) 40: (2,4)

%e 7: (1,1,1) 25: (1,3,1) 42: (2,2,2)

%e 8: (4) 26: (1,2,2) 44: (2,1,3)

%e 9: (3,1) 27: (1,2,1,1) 45: (2,1,2,1)

%e 11: (2,1,1) 28: (1,1,3) 47: (2,1,1,1,1)

%e 12: (1,3) 29: (1,1,2,1) 48: (1,5)

%e 14: (1,1,2) 30: (1,1,1,2) 49: (1,4,1)

%e 16: (5) 31: (1,1,1,1,1) 51: (1,3,1,1)

%e 17: (4,1) 32: (6) 52: (1,2,3)

%e 18: (3,2) 33: (5,1) 54: (1,2,1,2)

%e 19: (3,1,1) 34: (4,2) 56: (1,1,4)

%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[#]]!=0&]

%Y The version for Heinz numbers of partitions is A000037.

%Y These compositions are counted by A058622.

%Y These are the positions of terms != 0 in A124754.

%Y The complement (k = 0) is A344619.

%Y The positive (k > 0) version is A345917 (reverse: A345918).

%Y The negative (k < 0) version is A345919 (reverse: A345920).

%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 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, A000346, A008549, A025047, A032443, A034871, A114121, A163493, A236913, A344609, A344651, A345908.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 10 2021