A357621 - OEIS

%I #11 Oct 08 2022 08:34:20

%S 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,

%T 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,

%U 7,7,5,7,5,3,3,7,5,3,3,1,1,1,3,7,5,3,3,1

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

%C 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 - ...

%C 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.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

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

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

%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];

%t Table[halfats[stc[n]],{n,0,100}]

%Y See link for sequences related to standard compositions.

%Y The reverse version is A357622.

%Y The skew-alternating form is A357623, reverse A357624.

%Y Positions of zeros are A357625, reverse A357626.

%Y The version for prime indices is A357629.

%Y The version for Heinz numbers of partitions is A357633.

%Y A357637 counts partitions by half-alternating sum, skew A357638.

%Y A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.

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

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

%K sign

%O 0,3

%A _Gus Wiseman_, Oct 07 2022