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

37, 38, 41, 44, 50, 52, 69, 70, 81, 88, 98, 104, 133, 134, 137, 140, 145, 152, 161, 176, 194, 196, 200, 208, 261, 262, 265, 268, 274, 276, 289, 290, 296, 304, 321, 324, 328, 352, 386, 388, 400, 416, 517, 518, 521, 524, 529, 530, 532, 536, 545, 560, 577, 578

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..54.

Formula

These triples are counted by 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1).

Intersection of A014311 and A233564.

Example

The sequence together with the corresponding triples begins:
     37: (3,2,1)    140: (4,1,3)    289: (3,5,1)
     38: (3,1,2)    145: (3,4,1)    290: (3,4,2)
     41: (2,3,1)    152: (3,1,4)    296: (3,2,4)
     44: (2,1,3)    161: (2,5,1)    304: (3,1,5)
     50: (1,3,2)    176: (2,1,5)    321: (2,6,1)
     52: (1,2,3)    194: (1,5,2)    324: (2,4,3)
     69: (4,2,1)    196: (1,4,3)    328: (2,3,4)
     70: (4,1,2)    200: (1,3,4)    352: (2,1,6)
     81: (2,4,1)    208: (1,2,5)    386: (1,6,2)
     88: (2,1,4)    261: (6,2,1)    388: (1,5,3)
     98: (1,4,2)    262: (6,1,2)    400: (1,3,5)
    104: (1,2,4)    265: (5,3,1)    416: (1,2,6)
    133: (5,2,1)    268: (5,1,3)    517: (7,2,1)
    134: (5,1,2)    274: (4,3,2)    518: (7,1,2)
    137: (4,3,1)    276: (4,2,3)    521: (6,3,1)

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], Length[stc[#]]==3&&UnsameQ@@stc[#]&]

Crossrefs

6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts these compositions.

A007304 is an unordered version.

A014311 is the non-strict version.

A337461 counts the coprime case.

A000217(n - 2) counts 3-part compositions.

A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.

A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.

A014612 ranks 3-part partitions.

Cf. A000212, A220377, A307534, A337459, A337460, A337561, A337603, A337604.

Sequence in context: A043611 A296871 A346383 * A071887 A168143 A111043

Adjacent sequences: A337450 A337451 A337452 * A337454 A337455 A337456

Keyword

nonn

Author

Gus Wiseman, Sep 07 2020

Status

approved