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

0,3

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.

Links

Table of n, a(n) for n=0..84.

Gus Wiseman, Statistics, classes, and transformations of standard 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.

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

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

Sequence in context: A127714 A283763 A357622 * A220604 A331484 A046918

Adjacent sequences: A357618 A357619 A357620 * A357622 A357623 A357624

Keyword

sign

Author

Gus Wiseman, Oct 07 2022

Status

approved