cp's OEIS Frontend

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

%I A359756 #6 Jan 19 2023 11:10:50
%S A359756 0,3,6,7,13,14,15,27,29,30,31,55,59,61,62,63,111,119,123,125,126
%N A359756 First position of n in the sequence of zero-based weighted sums of standard compositions (A124757), if we start with position 0.
%C A359756 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.
%C A359756 The zero-based weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} (i-1)*y_i.
%C A359756 Is this sequence strictly increasing?
%H A359756 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 A359756 Appears to be the complement of A083329 in A089633.
%e A359756 The terms together with their standard compositions begin:
%e A359756     0: ()
%e A359756     3: (1,1)
%e A359756     6: (1,2)
%e A359756     7: (1,1,1)
%e A359756    13: (1,2,1)
%e A359756    14: (1,1,2)
%e A359756    15: (1,1,1,1)
%e A359756    27: (1,2,1,1)
%e A359756    29: (1,1,2,1)
%e A359756    30: (1,1,1,2)
%e A359756    31: (1,1,1,1,1)
%t A359756 nn=10;
%t A359756 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A359756 wts[y_]:=Sum[(i-1)*y[[i]],{i,Length[y]}];
%t A359756 seq=Table[wts[stc[n]],{n,0,2^(nn-1)}];
%t A359756 Table[Position[seq,k][[1,1]]-1,{k,0,nn}]
%Y A359756 The one-based version is A089633, for prime indices A359682.
%Y A359756 First index of n in A124757, reverse A231204.
%Y A359756 The version for prime indices is A359676, reverse A359681.
%Y A359756 A053632 counts compositions by zero-based weighted sum.
%Y A359756 A066099 lists standard compositions.
%Y A359756 A304818 gives weighted sums of prime indices, reverse A318283.
%Y A359756 A320387 counts multisets by weighted sum, zero-based A359678.
%Y A359756 Cf. A000120, A029931, A059893, A070939, A083329, A359043, A359674.
%K A359756 nonn,more
%O A359756 0,2
%A A359756 _Gus Wiseman_, Jan 17 2023