A342334 - OEIS

%I #14 Mar 12 2021 04:03:35

%S 1,1,1,2,3,4,6,11,16,23,35,54,82,125,193,294,447,680,1037,1580,2408,

%T 3676,5606,8544,13024,19860,30277,46155,70374,107300,163586,249397,

%U 380235,579705,883810,1347467,2054371,3132102,4775211,7280321,11099613,16922503,25800136,39335052,59970425,91431195

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

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

%e The a(1) = 1 through a(8) = 16 compositions:

%e   (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)

%e             (12)  (13)  (14)   (15)   (16)    (17)

%e                   (31)  (41)   (24)   (25)    (26)

%e                         (131)  (51)   (52)    (62)

%e                                (141)  (61)    (71)

%e                                (312)  (124)   (125)

%e                                       (151)   (152)

%e                                       (241)   (161)

%e                                       (313)   (251)

%e                                       (412)   (314)

%e                                       (1312)  (413)

%e                                               (512)

%e                                               (1241)

%e                                               (1313)

%e                                               (1412)

%e                                               (3131)

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

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

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

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

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

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

%Y A154402 counts partitions with adjacent parts x = 2y.

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

%Y A274199 counts compositions with adjacent parts x < 2y.

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

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

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

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

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

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

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Mar 10 2021

%E More terms from _Joerg Arndt_, Mar 12 2021