A342333 Number of compositions of n with all adjacent parts (x, y) satisfying x >= 2y or y >= 2x.

1, 1, 1, 3, 4, 5, 10, 18, 26, 42, 72, 114, 184, 305, 494, 799, 1305, 2123, 3446, 5611, 9134, 14851, 24162, 39314, 63945, 104025, 169238, 275305, 447863, 728592, 1185248, 1928143, 3136706, 5102743, 8301086, 13504175, 21968436, 35737995, 58138282, 94578751, 153859673

Offset: 0

Author

Gus Wiseman, Mar 10 2021

Keywords

• nonn

Comments

Either quotient x/y or y/x must be >= 2.

Examples

			The a(1) =  1 through a(7) = 18 compositions:
  (1)  (2)  (3)   (4)    (5)    (6)     (7)
            (12)  (13)   (14)   (15)    (16)
            (21)  (31)   (41)   (24)    (25)
                  (121)  (131)  (42)    (52)
                         (212)  (51)    (61)
                                (141)   (124)
                                (213)   (142)
                                (312)   (151)
                                (1212)  (214)
                                (2121)  (241)
                                        (313)
                                        (412)
                                        (421)
                                        (1213)
                                        (1312)
                                        (2131)
                                        (3121)
                                        (12121)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

The unordered version (partitions) is A000929.

Reversing operators and changing 'or' into 'and' gives A224957 (strict: A342342).

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

The version allowing partial equality is A342334.

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 parts x <= 2y (strict: A342095).

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

A342098 counts partitions with adjacent parts x > 2y.

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.

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

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

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, j), j=
      `if`(i=0, 1..n, {$1..min(n, iquo(i, 2)), $(2*i)..n})))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..42);  # Alois P. Heinz, May 24 2021

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]>=2*#[[i-1]]||#[[i-1]]>=2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, Sum[b[n-j, j], {j, 1, n}], Sum[b[n-j, j], {j, Range[Min[n, Quotient[i, 2]]]~Union~Range[2i, n]}]]];
a[n_] := b[n, 0];
a /@ Range[0, 42] (* Jean-François Alcover, Jun 10 2021, after_Alois P. Heinz_ *)

Extensions

More terms from Joerg Arndt, Mar 12 2021