A342342 Number of strict compositions of n with all adjacent parts (x, y) satisfying x <= 2y and y <= 2x.

1, 1, 1, 3, 1, 3, 5, 5, 3, 11, 9, 11, 17, 15, 29, 39, 31, 39, 65, 57, 107, 127, 149, 155, 187, 265, 293, 419, 523, 571, 781, 763, 941, 1371, 1387, 2125, 2383, 2775, 3243, 4189, 4555, 5349, 7241, 7997, 10591, 13171, 14581, 17213, 20253, 25177, 27701, 34317

Offset: 0

Author

Gus Wiseman, Mar 12 2021

Keywords

• nonn

Comments

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

Examples

			The a(1) = 1 through a(12) = 17 strict compositions (A = 10, B = 11, C = 12):
  1   2   3    4   5    6     7     8    9     A      B      C
          12       23   24    34    35   36    46     47     48
          21       32   42    43    53   45    64     56     57
                        123   124        54    235    65     75
                        321   421        63    532    74     84
                                         234   1234   236    246
                                         243   1243   245    345
                                         324   3421   542    354
                                         342   4321   632    435
                                         423          1235   453
                                         432          5321   534
                                                             543
                                                             642
                                                             1236
                                                             1245
                                                             5421
                                                             6321
		

Crossrefs

The non-strict version is A224957.

The case with strict relations is A342341 (non-strict: A342330).

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).

A342096 counts partitions without adjacent x >= 2y (strict: A342097).

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, A342340.

Programs

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}]

Extensions

a(40)-a(51) from Alois P. Heinz, May 24 2021