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A359756
First position of n in the sequence of zero-based weighted sums of standard compositions (A124757), if we start with position 0.
7
0, 3, 6, 7, 13, 14, 15, 27, 29, 30, 31, 55, 59, 61, 62, 63, 111, 119, 123, 125, 126
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?
FORMULA
Appears to be the complement of A083329 in A089633.
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
The one-based version is A089633, for prime indices A359682.
First index of n in A124757, reverse A231204.
The version for prime indices is A359676, reverse A359681.
A053632 counts compositions by zero-based weighted sum.
A066099 lists standard compositions.
A304818 gives weighted sums of prime indices, reverse A318283.
A320387 counts multisets by weighted sum, zero-based A359678.
Sequence in context: A294231 A280873 A293437 * A350943 A282354 A088146
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jan 17 2023
STATUS
approved