%I #5 Sep 17 2020 20:33:59
%S 37,38,41,44,50,52,69,70,81,88,98,104,133,134,137,140,145,152,161,176,
%T 194,196,200,208,261,262,265,268,274,276,289,290,296,304,321,324,328,
%U 352,386,388,400,416,517,518,521,524,529,530,532,536,545,560,577,578
%N Numbers k such that the k-th composition in standard order is an ordered triple of distinct positive integers.
%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.
%F These triples are counted by 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1).
%F Intersection of A014311 and A233564.
%e The sequence together with the corresponding triples begins:
%e 37: (3,2,1) 140: (4,1,3) 289: (3,5,1)
%e 38: (3,1,2) 145: (3,4,1) 290: (3,4,2)
%e 41: (2,3,1) 152: (3,1,4) 296: (3,2,4)
%e 44: (2,1,3) 161: (2,5,1) 304: (3,1,5)
%e 50: (1,3,2) 176: (2,1,5) 321: (2,6,1)
%e 52: (1,2,3) 194: (1,5,2) 324: (2,4,3)
%e 69: (4,2,1) 196: (1,4,3) 328: (2,3,4)
%e 70: (4,1,2) 200: (1,3,4) 352: (2,1,6)
%e 81: (2,4,1) 208: (1,2,5) 386: (1,6,2)
%e 88: (2,1,4) 261: (6,2,1) 388: (1,5,3)
%e 98: (1,4,2) 262: (6,1,2) 400: (1,3,5)
%e 104: (1,2,4) 265: (5,3,1) 416: (1,2,6)
%e 133: (5,2,1) 268: (5,1,3) 517: (7,2,1)
%e 134: (5,1,2) 274: (4,3,2) 518: (7,1,2)
%e 137: (4,3,1) 276: (4,2,3) 521: (6,3,1)
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Select[Range[0,100],Length[stc[#]]==3&&UnsameQ@@stc[#]&]
%Y 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts these compositions.
%Y A007304 is an unordered version.
%Y A014311 is the non-strict version.
%Y A337461 counts the coprime case.
%Y A000217(n - 2) counts 3-part compositions.
%Y A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
%Y A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.
%Y A014612 ranks 3-part partitions.
%Y Cf. A000212, A220377, A307534, A337459, A337460, A337561, A337603, A337604.
%K nonn
%O 1,1
%A _Gus Wiseman_, Sep 07 2020