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A357623
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Skew-alternating sum of the n-th composition in standard order.
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23
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0, 1, 2, 0, 3, 1, -1, -1, 4, 2, 0, 0, -2, -2, -2, 0, 5, 3, 1, 1, -1, -1, -1, 1, -3, -3, -3, -1, -3, -1, 1, 1, 6, 4, 2, 2, 0, 0, 0, 2, -2, -2, -2, 0, -2, 0, 2, 2, -4, -4, -4, -2, -4, -2, 0, 0, -4, -2, 0, 0, 2, 2, 2, 0, 7, 5, 3, 3, 1, 1, 1, 3, -1, -1, -1, 1, -1
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OFFSET
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0,3
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COMMENTS
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We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....
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 358-th composition is (2,1,3,1,2) so a(358) = 2 - 1 - 3 + 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;
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[skats[stc[n]], {n, 0, 100}]
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CROSSREFS
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See link for sequences related to standard compositions.
Positions of positive firsts appear to be A029744.
The version for prime indices is A357630.
The version for Heinz numbers of partitions is A357634.
A124754 gives alternating sum of standard compositions, reverse A344618.
Cf. A001700, A001511, A053251, A344619, A357136, A357182, A357183, A357184, A357185, A357625, A357626.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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