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A357621
Half-alternating sum of the n-th composition in standard order.
28
0, 1, 2, 2, 3, 3, 3, 1, 4, 4, 4, 2, 4, 2, 0, 0, 5, 5, 5, 3, 5, 3, 1, 1, 5, 3, 1, 1, -1, -1, -1, 1, 6, 6, 6, 4, 6, 4, 2, 2, 6, 4, 2, 2, 0, 0, 0, 2, 6, 4, 2, 2, 0, 0, 0, 2, -2, -2, -2, 0, -2, 0, 2, 2, 7, 7, 7, 5, 7, 5, 3, 3, 7, 5, 3, 3, 1, 1, 1, 3, 7, 5, 3, 3, 1
OFFSET
0,3
COMMENTS
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.
FORMULA
Positions of first appearances are powers of 2 and even powers of 2 times 7, or A029746 without 7.
EXAMPLE
The 358-th composition is (2,1,3,1,2) so a(358) = 2 + 1 - 3 - 1 + 2 = 1.
MATHEMATICA
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[stc[n]], {n, 0, 100}]
CROSSREFS
See link for sequences related to standard compositions.
The reverse version is A357622.
The skew-alternating form is A357623, reverse A357624.
Positions of zeros are A357625, reverse A357626.
The version for prime indices is A357629.
The version for Heinz numbers of partitions is A357633.
A357637 counts partitions by half-alternating sum, skew A357638.
A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
Sequence in context: A127714 A283763 A357622 * A220604 A331484 A046918
KEYWORD
sign
AUTHOR
Gus Wiseman, Oct 07 2022
STATUS
approved