A342334 Number of compositions of n with all adjacent parts (x, y) satisfying x >= 2y or y > 2x.

1, 1, 1, 2, 3, 4, 6, 11, 16, 23, 35, 54, 82, 125, 193, 294, 447, 680, 1037, 1580, 2408, 3676, 5606, 8544, 13024, 19860, 30277, 46155, 70374, 107300, 163586, 249397, 380235, 579705, 883810, 1347467, 2054371, 3132102, 4775211, 7280321, 11099613, 16922503, 25800136, 39335052, 59970425, 91431195

Offset: 0

Author

Gus Wiseman, Mar 10 2021

Keywords

• nonn

Comments

Also the number of compositions of n with all adjacent parts (x, y) satisfying x > 2y or y >= 2x.

Examples

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (12)  (13)  (14)   (15)   (16)    (17)
                  (31)  (41)   (24)   (25)    (26)
                        (131)  (51)   (52)    (62)
                               (141)  (61)    (71)
                               (312)  (124)   (125)
                                      (151)   (152)
                                      (241)   (161)
                                      (313)   (251)
                                      (412)   (314)
                                      (1312)  (413)
                                              (512)
                                              (1241)
                                              (1313)
                                              (1412)
                                              (3131)
		

Crossrefs

The unordered version (partitions) is A342098 or A000929 (multisets).

The version not allowing equality (i.e., strict relations) is A342332.

The version allowing equality (i.e., non-strict relations) is A342333.

Reversing operators and changing 'or' into 'and' gives A342338.

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

A154402 counts partitions with adjacent parts x = 2y.

A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).

A274199 counts compositions with adjacent parts x < 2y.

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

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

A342330 counts compositions with x < 2y and y < 2x (strict: A342341).

A342331 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.

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

Programs

Mathematica

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

Extensions

More terms from Joerg Arndt, Mar 12 2021