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A337453 Numbers k such that the k-th composition in standard order is an ordered triple of distinct positive integers.

%I A337453 #5 Sep 17 2020 20:33:59
%S A337453 37,38,41,44,50,52,69,70,81,88,98,104,133,134,137,140,145,152,161,176,
%T A337453 194,196,200,208,261,262,265,268,274,276,289,290,296,304,321,324,328,
%U A337453 352,386,388,400,416,517,518,521,524,529,530,532,536,545,560,577,578
%N A337453 Numbers k such that the k-th composition in standard order is an ordered triple of distinct positive integers.
%C A337453 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 A337453 These triples are counted by 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1).
%F A337453 Intersection of A014311 and A233564.
%e A337453 The sequence together with the corresponding triples begins:
%e A337453      37: (3,2,1)    140: (4,1,3)    289: (3,5,1)
%e A337453      38: (3,1,2)    145: (3,4,1)    290: (3,4,2)
%e A337453      41: (2,3,1)    152: (3,1,4)    296: (3,2,4)
%e A337453      44: (2,1,3)    161: (2,5,1)    304: (3,1,5)
%e A337453      50: (1,3,2)    176: (2,1,5)    321: (2,6,1)
%e A337453      52: (1,2,3)    194: (1,5,2)    324: (2,4,3)
%e A337453      69: (4,2,1)    196: (1,4,3)    328: (2,3,4)
%e A337453      70: (4,1,2)    200: (1,3,4)    352: (2,1,6)
%e A337453      81: (2,4,1)    208: (1,2,5)    386: (1,6,2)
%e A337453      88: (2,1,4)    261: (6,2,1)    388: (1,5,3)
%e A337453      98: (1,4,2)    262: (6,1,2)    400: (1,3,5)
%e A337453     104: (1,2,4)    265: (5,3,1)    416: (1,2,6)
%e A337453     133: (5,2,1)    268: (5,1,3)    517: (7,2,1)
%e A337453     134: (5,1,2)    274: (4,3,2)    518: (7,1,2)
%e A337453     137: (4,3,1)    276: (4,2,3)    521: (6,3,1)
%t A337453 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A337453 Select[Range[0,100],Length[stc[#]]==3&&UnsameQ@@stc[#]&]
%Y A337453 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts these compositions.
%Y A337453 A007304 is an unordered version.
%Y A337453 A014311 is the non-strict version.
%Y A337453 A337461 counts the coprime case.
%Y A337453 A000217(n - 2) counts 3-part compositions.
%Y A337453 A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
%Y A337453 A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.
%Y A337453 A014612 ranks 3-part partitions.
%Y A337453 Cf. A000212, A220377, A307534, A337459, A337460, A337561, A337603, A337604.
%K A337453 nonn
%O A337453 1,1
%A A337453 _Gus Wiseman_, Sep 07 2020