A348382 Number of compositions of n that are not a twin (x,x) but have adjacent equal parts.

0, 0, 0, 1, 3, 9, 17, 41, 88, 185, 387, 810, 1669, 3435, 7039, 14360, 29225, 59347, 120228, 243166, 491085, 990446, 1995409, 4016259, 8076959, 16231746, 32599773, 65437945, 131293191, 263316897, 527912139, 1058061751, 2120039884, 4246934012, 8505864639

Offset: 0

Author

Gus Wiseman, Nov 05 2021

Keywords

• nonn

Comments

A composition with no adjacent equal parts is also called a Carlitz composition, so these are non-twin, non-Carlitz compositions.

Examples

			The a(3) = 1 through a(6) = 17 compositions:
  (111)  (112)   (113)    (114)
         (211)   (122)    (222)
         (1111)  (221)    (411)
                 (311)    (1113)
                 (1112)   (1122)
                 (1121)   (1131)
                 (1211)   (1221)
                 (2111)   (1311)
                 (11111)  (2112)
                          (2211)
                          (3111)
                          (11112)
                          (11121)
                          (11211)
                          (12111)
                          (21111)
                          (111111)
		

Links

• A. Knopfmacher and H. Prodinger, On Carlitz Compositions, Europ. J. Combinatorics (1998) 19, 579-589.
• Wikipedia, Composition (combinatorics)

Crossrefs

Allowing twins gives A261983, complement A003242.

The non-alternating case is A348377, difference A345195.

These compositions are ranked by A348612 \ A007582.

A001250 counts alternating permutations, complement A348615.

A007582 ranks twin compositions.

A011782 counts compositions, strict A032020.

A025047 counts alternating or wiggly compositions, complement A345192.

A051049 counts non-twin compositions, complement A000035(n+1).

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

Cf. A000070, A005649, A059841, A106356, A238279, A333755, A344604, A344614, A344740, A348381.

Programs

Mathematica

nn=15;CoefficientList[Series[1+x/(1-2x)-x^2/(1-x^2)-1/(1-Sum[x^k/(1+x^k),{k,1,nn}]),{x,0,nn}],x]

Formula

For n > 0, a(n) = A261983(n) - A059841(n).

O.g.f.: 1 + x/(1-2x) - x^2/(1-x^2) - 1/(1 - Sum_{k>0} x^k/(1+x^k)).