A357706 - OEIS

%I #6 Oct 13 2022 12:28:23

%S 0,15,45,54,59,153,170,179,204,213,230,235,247,255,561,594,611,660,

%T 677,710,715,727,735,750,765,792,809,842,851,871,879,894,908,917,934,

%U 939,951,959,973,982,987,1005,1014,1019

%N Numbers k such that the k-th composition in standard order has half-alternating sum and skew-alternating sum both 0.

%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 - ..., and the skew-alternating sum 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.

%F Intersection of A357625 and A357627.

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

%t Select[Range[0,1000],halfats[stc[#]]==0&&skats[stc[#]]==0&]

%Y For partitions and half only (or both): A000583, counted by A035363.

%Y These compositions are counted by A228248.

%Y For half-alternating only: A357625, reverse A357626, counted by A357641.

%Y For skew-alternating only: A357627, reverse A357628, counted by A001700.

%Y For reversed partitions and half only: A357631, counted by A357639.

%Y For reversed partitions and skew only A357632, counted by A357640.

%Y For partitions and skew only: A357636, counted by A035594.

%Y Cf. A001511, A004006, A088218, A357136, A357182, A357642.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 13 2022