OFFSET
0,2
COMMENTS
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.
The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.
Is this sequence strictly increasing?
LINKS
EXAMPLE
The terms together with their standard compositions begin:
0: ()
3: (1,1)
6: (1,2)
7: (1,1,1)
13: (1,2,1)
14: (1,1,2)
15: (1,1,1,1)
27: (1,2,1,1)
29: (1,1,2,1)
30: (1,1,1,2)
31: (1,1,1,1,1)
MATHEMATICA
nn=10;
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
wts[y_]:=Sum[(i-1)*y[[i]], {i, Length[y]}];
seq=Table[wts[stc[n]], {n, 0, 2^(nn-1)}];
Table[Position[seq, k][[1, 1]]-1, {k, 0, nn}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jan 17 2023
STATUS
approved