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A345920
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Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum < 0.
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26
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5, 9, 17, 18, 23, 25, 29, 33, 34, 39, 45, 49, 57, 65, 66, 68, 71, 75, 77, 78, 81, 85, 89, 90, 95, 97, 98, 103, 105, 109, 113, 114, 119, 121, 125, 129, 130, 132, 135, 139, 141, 142, 149, 153, 154, 159, 161, 169, 177, 178, 183, 189, 193, 194, 199, 205, 209, 217
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OFFSET
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1,1
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COMMENTS
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The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) 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 the corresponding compositions:
5: (2,1) 68: (4,3)
9: (3,1) 71: (4,1,1,1)
17: (4,1) 75: (3,2,1,1)
18: (3,2) 77: (3,1,2,1)
23: (2,1,1,1) 78: (3,1,1,2)
25: (1,3,1) 81: (2,4,1)
29: (1,1,2,1) 85: (2,2,2,1)
33: (5,1) 89: (2,1,3,1)
34: (4,2) 90: (2,1,2,2)
39: (3,1,1,1) 95: (2,1,1,1,1,1)
45: (2,1,2,1) 97: (1,5,1)
49: (1,4,1) 98: (1,4,2)
57: (1,1,3,1) 103: (1,3,1,1,1)
65: (6,1) 105: (1,2,3,1)
66: (5,2) 109: (1,2,1,2,1)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Select[Range[0, 100], sats[stc[#]]<0&]
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CROSSREFS
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The version for prime indices is {}.
The version for Heinz numbers of partitions is A119899.
These are the positions of terms < 0 in A344618.
The weak (k <= 0) version is A345916.
The opposite (k > 0) version is A345918.
The version for unreversed alternating sum is A345919.
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).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
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, A000346, A025047, A028260, A032443, A034871, A106356, A114121, A163493, A344608, A344610, A344611, A345908.
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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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