A342341 Number of strict compositions of n with all adjacent parts (x, y) satisfying x < 2y and y < 2x.

1, 1, 1, 1, 1, 3, 1, 3, 3, 5, 5, 5, 9, 7, 13, 15, 17, 19, 29, 31, 39, 43, 63, 59, 75, 121, 119, 169, 167, 199, 279, 305, 343, 479, 537, 733, 789, 883, 1057, 1421, 1545, 1831, 2409, 2577, 3343, 4001, 4657, 5131, 6065, 7755, 8841, 10473, 12995, 14659, 17671, 20619, 25157, 28255, 33131, 38265, 47699, 53171, 62611, 80005, 88519, 105937, 119989

Offset

0,6

Comments

Each quotient of adjacent parts is between 1/2 and 2 exclusive.

Links

Bert Dobbelaere, Table of n, a(n) for n = 0..100

Example

The a(1) = 1 through a(17) = 17 compositions (A..G = 10..16):
  1  2  3  4  5   6  7   8   9    A    B   C    D    E     F     G
              23     34  35  45   46   47  57   58   59    69    6A
              32     43  53  54   64   56  75   67   68    78    79
                             234  235  65  345  76   86    87    97
                             432  532  74  354  85   95    96    A6
                                           435  346  347   357   358
                                           453  643  356   456   457
                                           534       653   465   475
                                           543       743   546   547
                                                     2345  564   574
                                                     2354  645   745
                                                     4532  654   754
                                                     5432  753   853
                                                           2346  2347
                                                           6432  2356
                                                                 6532
                                                                 7432

Mathematica

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], UnsameQ@@#&], And@@Table[#[[i]]<2*#[[i-1]]&&#[[i-1]]<2*#[[i]], {i, 2, Length[#]}]&]], {n, 0, 15}]

Crossrefs

The unordered version (partitions) is A342097 (non-strict: A342096).

The non-strict version is A342330.

The version allowing equality is A342342 (non-strict: A224957).

A000929 counts partitions with adjacent parts x >= 2y.

A002843 counts compositions with adjacent parts x <= 2y.

A154402 counts partitions with adjacent parts x = 2y.

A274199 counts compositions with adjacent parts x < 2y.

A342094 counts partitions with adjacent x <= 2y (strict: A342095).

A342098 counts partitions with adjacent parts x > 2y.

A342331 counts compositions with adjacent parts x = 2y or y = 2x.

A342332 counts compositions with adjacent parts x > 2y or y > 2x.

A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.

A342335 counts compositions with adjacent parts x >= 2y or y = 2x.

A342337 counts partitions with adjacent parts x = y or x = 2y.

A342338 counts compositions with adjacent parts x < 2y and y <= 2x.

Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342334, A342336, A342339, A342340.

Sequence in context: A085723 A186422 A165027 * A078555 A259286 A087891

Adjacent sequences: A342338 A342339 A342340 * A342342 A342343 A342344

Keyword

nonn

Author

Gus Wiseman, Mar 12 2021

Extensions

More terms from Bert Dobbelaere, Mar 19 2021

Status

approved