A359756 First position of n in the sequence of zero-based weighted sums of standard compositions (A124757), if we start with position 0.

0, 3, 6, 7, 13, 14, 15, 27, 29, 30, 31, 55, 59, 61, 62, 63, 111, 119, 123, 125, 126

Offset: 0

Author

Gus Wiseman, Jan 17 2023

Keywords

• nonn
• more

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?

Examples

			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)
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

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.

Cf. A000120, A029931, A059893, A070939, A083329, A359043, A359674.

Programs

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}]

Formula

Appears to be the complement of A083329 in A089633.