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