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A357622
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Half-alternating sum of the reversed n-th composition in standard order.
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13
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0, 1, 2, 2, 3, 3, 3, 1, 4, 4, 4, 0, 4, 2, 2, 0, 5, 5, 5, -1, 5, 1, 1, -1, 5, 3, 3, -1, 3, 1, 1, 1, 6, 6, 6, -2, 6, 0, 0, -2, 6, 2, 2, -2, 2, 0, 0, 2, 6, 4, 4, -2, 4, 0, 0, 0, 4, 2, 2, 0, 2, 2, 2, 2, 7, 7, 7, -3, 7, -1, -1, -3, 7, 1, 1, -3, 1, -1, -1, 3, 7, 3
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OFFSET
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0,3
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COMMENTS
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We define the half-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 357-th composition is (2,1,3,2,1) so a(357) = 1 + 2 - 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;
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Table[halfats[Reverse[stc[n]]], {n, 0, 100}]
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CROSSREFS
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See link for sequences related to standard compositions.
This is the reverse version of A357621.
The version for prime indices is A357629.
The version for Heinz numbers of partitions is A357633.
A124754 gives alternating sum of standard compositions, reverse A344618.
Cf. A001511, A053251, A357136, A357182, A357183, A357184, A357185, A357627, A357628, A357631, A357635.
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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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