A348612 Numbers k such that the k-th composition in standard order is not an anti-run, i.e., has adjacent equal parts.

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 46, 47, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 75, 78, 79, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 99, 100, 103, 106, 107, 110, 111, 112, 113, 114, 115, 116

Offset

1,1

Comments

First differs from A345168 in lacking 37, corresponding to the composition (3,2,1).

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.

Links

Table of n, a(n) for n=1..63.

Example

The terms and corresponding standard compositions begin:
     3: (1,1)          35: (4,1,1)        61: (1,1,1,2,1)
     7: (1,1,1)        36: (3,3)          62: (1,1,1,1,2)
    10: (2,2)          39: (3,1,1,1)      63: (1,1,1,1,1,1)
    11: (2,1,1)        42: (2,2,2)        67: (5,1,1)
    14: (1,1,2)        43: (2,2,1,1)      71: (4,1,1,1)
    15: (1,1,1,1)      46: (2,1,1,2)      73: (3,3,1)
    19: (3,1,1)        47: (2,1,1,1,1)    74: (3,2,2)
    21: (2,2,1)        51: (1,3,1,1)      75: (3,2,1,1)
    23: (2,1,1,1)      53: (1,2,2,1)      78: (3,1,1,2)
    26: (1,2,2)        55: (1,2,1,1,1)    79: (3,1,1,1,1)
    27: (1,2,1,1)      56: (1,1,4)        83: (2,3,1,1)
    28: (1,1,3)        57: (1,1,3,1)      84: (2,2,3)
    29: (1,1,2,1)      58: (1,1,2,2)      85: (2,2,2,1)
    30: (1,1,1,2)      59: (1,1,2,1,1)    86: (2,2,1,2)
    31: (1,1,1,1,1)    60: (1,1,1,3)      87: (2,2,1,1,1)

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[100], MatchQ[stc[#], {___, x_, x_, ___}]&]

Crossrefs

Constant run compositions are counted by A000005, ranked by A272919.

Counting these compositions by sum and length gives A131044.

These compositions are counted by A261983.

The complement is A333489, counted by A003242.

The non-alternating case is A345168, complement A345167.

A011782 counts compositions, strict A032020.

A238279 counts compositions by sum and number of maximal runs.

A274174 counts compositions with equal parts contiguous.

A336107 counts non-anti-run permutations of prime factors.

A345195 counts non-alternating anti-runs, ranked by A345169.

For compositions in standard order (rows of A066099):

- Length is A000120.

- Sum is A070939

- Maximal runs are counted by A124767.

- Strict compositions are ranked by A233564.

- Maximal anti-runs are counted by A333381.

- Runs-resistance is A333628.

Cf. A029931, A048793, A106356, A114901, A167606, A178470, A228351, A244164, A262046, A335452, A335464.

Sequence in context: A335488 A285036 A345168 * A349799 A360402 A188081

Adjacent sequences: A348609 A348610 A348611 * A348613 A348614 A348615

Keyword

nonn

Author

Gus Wiseman, Nov 03 2021

Status

approved