A357621 Half-alternating sum of the n-th composition in standard order.

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

Author

Gus Wiseman, Oct 07 2022

Keywords

• sign

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.

Examples

			The 358-th composition is (2,1,3,1,2) so a(358) = 2 + 1 - 3 - 1 + 2 = 1.
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

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.

Cf. A001511, A053251, A357136, A357182, A357183, A357184, A357185.

Cf. A357627, A357628, A357630, A357631, A357634, A357635, A357640.

Programs

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}]

Formula

Positions of first appearances are powers of 2 and even powers of 2 times 7, or A029746 without 7.