A357708 - OEIS

%I #6 Oct 15 2022 08:10:51

%S 3,10,11,13,14,36,37,38,39,41,44,50,51,52,57,60,136,137,138,139,140,

%T 141,142,143,145,152,162,163,168,177,184,196,197,198,199,200,209,216,

%U 226,227,232,241,248,528,529,530,531,532,533,534,535,536,537,538,539

%N Numbers k such that the k-th composition in standard order has sum equal to twice its maximum part.

%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.

%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>

%e The terms and corresponding compositions begin:

%e     3: (1,1)

%e    10: (2,2)

%e    11: (2,1,1)

%e    13: (1,2,1)

%e    14: (1,1,2)

%e    36: (3,3)

%e    37: (3,2,1)

%e    38: (3,1,2)

%e    39: (3,1,1,1)

%e    41: (2,3,1)

%e    44: (2,1,3)

%e    50: (1,3,2)

%e    51: (1,3,1,1)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,1000],Max@@stc[#]==Total[stc[#]]/2&]

%Y See link for sequences related to standard compositions.

%Y A066311 lists gapless numbers.

%Y A124767 counts runs in standard compositions.

%Y A333766 gives maximal part of standard compositions, minimal A333768.

%Y A356844 ranks compositions with at least one 1.

%Y Cf. A000120, A001511, A003754, A029931, A329395.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 14 2022