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A345910
Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -1.
31
6, 20, 25, 27, 30, 72, 81, 83, 86, 92, 98, 101, 103, 106, 109, 111, 116, 121, 123, 126, 272, 289, 291, 294, 300, 312, 322, 325, 327, 330, 333, 335, 340, 345, 347, 350, 360, 369, 371, 374, 380, 388, 393, 395, 398, 402, 405, 407, 410, 413, 415, 420, 425, 427
OFFSET
1,1
COMMENTS
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
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.
EXAMPLE
The sequence of terms together with the corresponding compositions begins:
6: (1,2)
20: (2,3)
25: (1,3,1)
27: (1,2,1,1)
30: (1,1,1,2)
72: (3,4)
81: (2,4,1)
83: (2,3,1,1)
86: (2,2,1,2)
92: (2,1,1,3)
98: (1,4,2)
101: (1,3,2,1)
103: (1,3,1,1,1)
106: (1,2,2,2)
109: (1,2,1,2,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Select[Range[0, 100], ats[stc[#]]==-1&]
CROSSREFS
These compositions are counted by A001791.
A version using runs of binary digits is A031444.
These are the positions of -1's in A124754.
The opposite (positive 1) version is A345909.
The reverse version is A345912.
The version for alternating sum of prime indices is A345959.
Standard compositions: A000120, A066099, A070939, A124754, A228351, A344618.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions of 2n+1 with alternating sum 1, ranked by A001105.
A011782 counts compositions.
A097805 counts compositions by sum and alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0: counted by A088218, ranked by A344619/A344619.
- k = 1: counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2: counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0: counted by A027306, ranked by A345917/A345918.
- k < 0: counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd: counted by A000302, ranked by A053738/A053738.
Sequence in context: A341298 A308324 A044970 * A316291 A090502 A324649
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 01 2021
STATUS
approved