A342332 Number of compositions of n with all adjacent parts (x, y) satisfying x > 2y or y > 2x.

1, 1, 1, 1, 3, 4, 4, 7, 12, 17, 23, 34, 51, 75, 111, 164, 239, 350, 520, 767, 1123, 1652, 2439, 3587, 5263, 7745, 11411, 16789, 24695, 36347, 53489, 78686, 115779, 170390, 250711, 368866, 542783, 798713, 1175208, 1729189, 2544462, 3744077, 5509068, 8106165, 11927785, 17550956, 25824938, 37999743, 55914293, 82274088, 121060721

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2500

Example

The a(1) =  1 through a(9) = 17 compositions:
  (1)  (2)  (3)  (4)   (5)    (6)    (7)    (8)     (9)
                 (13)  (14)   (15)   (16)   (17)    (18)
                 (31)  (41)   (51)   (25)   (26)    (27)
                       (131)  (141)  (52)   (62)    (72)
                                     (61)   (71)    (81)
                                     (151)  (152)   (162)
                                     (313)  (161)   (171)
                                            (251)   (252)
                                            (314)   (261)
                                            (413)   (315)
                                            (1313)  (414)
                                            (3131)  (513)
                                                    (1314)
                                                    (1413)
                                                    (3141)
                                                    (4131)
                                                    (13131)

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, j),
      j=select(x-> i=0 or x>2*i or i>2*x , {$1..n})))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # 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, Sum[b[n - j, j], {j, Select[Range[n], i == 0 || # > 2 i || i > 2 # &]}]];
a[n_] := b[n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)

Crossrefs

The unordered version (partitions) is A342098.

Reversing operators and changing 'or' into 'and' gives A342330 (strict: A342341).

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

The version allowing partial equality is counted by A342334.

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

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).

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.

Sequence in context: A231343 A121924 A241740 * A225738 A186679 A275742

Adjacent sequences: A342329 A342330 A342331 * A342333 A342334 A342335

Keyword

nonn

Author

Gus Wiseman, Mar 10 2021

Extensions

More terms from Joerg Arndt, Mar 12 2021

Status

approved