%I #10 May 17 2022 17:47:30
%S 0,1,10,21,26,43,58,107,117,174,186,292,314,346,348,349,373,430,442,
%T 570,585,586,629,676,696,697,804,826,860,861,885,1082,1141,1173,1210,
%U 1338,1387,1392,1393,1394,1396,1594,1653,1700,1720,1721,1882,2106,2165,2186
%N Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.
%C First differs from A353402 (the non-consecutive version) in lacking 53.
%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.
%e The initial terms, their binary expansions, and the corresponding standard compositions:
%e 0: 0 ()
%e 1: 1 (1)
%e 10: 1010 (2,2)
%e 21: 10101 (2,2,1)
%e 26: 11010 (1,2,2)
%e 43: 101011 (2,2,1,1)
%e 58: 111010 (1,1,2,2)
%e 107: 1101011 (1,2,2,1,1)
%e 117: 1110101 (1,1,2,2,1)
%e 174: 10101110 (2,2,1,1,2)
%e 186: 10111010 (2,1,1,2,2)
%e 292: 100100100 (3,3,3)
%e 314: 100111010 (3,1,1,2,2)
%e 346: 101011010 (2,2,1,2,2)
%e 348: 101011100 (2,2,1,1,3)
%e 349: 101011101 (2,2,1,1,2,1)
%e 373: 101110101 (2,1,1,2,2,1)
%e 430: 110101110 (1,2,2,1,1,2)
%e 442: 110111010 (1,2,1,1,2,2)
%t stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t rorQ[y_]:=Length[y]==0||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]];
%t Select[Range[0,10000],rorQ[stc[#]]&]
%Y These compositions are counted by A353392.
%Y This is the consecutive case of A353402, counted by A353390.
%Y The non-consecutive recursive version is A353431, counted by A353391.
%Y The recursive version is A353696, counted by A353430.
%Y A005811 counts runs in binary expansion.
%Y A011782 counts compositions.
%Y A066099 lists compositions in standard order, rev A228351, run-lens A333769.
%Y A329738 counts uniform compositions, partitions A047966.
%Y Statistics of standard compositions:
%Y - Length is A000120, sum A070939.
%Y - Runs are counted by A124767, distinct A351014.
%Y - Subsequences are counted by A334299, contiguous A124770/A124771.
%Y - Runs-resistance is A333628.
%Y Classes of standard compositions:
%Y - Partitions are A114994, strict A333255, rev A225620, strict rev A333256.
%Y - Runs are A272919, counted by A000005.
%Y - Golomb rulers are A333222, counted by A169942.
%Y - Anti-runs are A333489, counted by A003242.
%Y Cf. A044813, A165413, A181819, A318928, A325702, A325705, A325755, A333224, A333755, A353389, A353393, A353403.
%K nonn
%O 1,3
%A _Gus Wiseman_, May 16 2022