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A345925
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Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 2.
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28
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2, 9, 11, 14, 34, 37, 39, 42, 45, 47, 52, 57, 59, 62, 132, 137, 139, 142, 146, 149, 151, 154, 157, 159, 164, 169, 171, 174, 178, 181, 183, 186, 189, 191, 200, 209, 211, 214, 220, 226, 229, 231, 234, 237, 239, 244, 249, 251, 254, 520, 529, 531, 534, 540, 546
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The initial terms and corresponding compositions:
2: (2) 137: (4,3,1)
9: (3,1) 139: (4,2,1,1)
11: (2,1,1) 142: (4,1,1,2)
14: (1,1,2) 146: (3,3,2)
34: (4,2) 149: (3,2,2,1)
37: (3,2,1) 151: (3,2,1,1,1)
39: (3,1,1,1) 154: (3,1,2,2)
42: (2,2,2) 157: (3,1,1,2,1)
45: (2,1,2,1) 159: (3,1,1,1,1,1)
47: (2,1,1,1,1) 164: (2,3,3)
52: (1,2,3) 169: (2,2,3,1)
57: (1,1,3,1) 171: (2,2,2,1,1)
59: (1,1,2,1,1) 174: (2,2,1,1,2)
62: (1,1,1,1,2) 178: (2,1,3,2)
132: (5,3) 181: (2,1,2,2,1)
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MATHEMATICA
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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[#]]==2&]
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CROSSREFS
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These compositions are counted by A088218.
These are the positions of 2's in A124754.
The case of partitions of 2n is A344741.
The version for reverse-alternating sum is A345922.
The opposite (negative 2) version is A345924.
The version for Heinz numbers of partitions is A345960 (reverse: A345961).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
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:
Cf. A000070, A000097, A025047, A114121, A163493, A238279, A239830, A344607, A344608, A344609, A344651, A344743.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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