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A348382 Number of compositions of n that are not a twin (x,x) but have adjacent equal parts. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

LINKS

Table of n, a(n) for n=0..34.

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

Wikipedia, Composition (combinatorics)

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)).

EXAMPLE

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)

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]

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.

Sequence in context: A011755 A262466 A128301 * A176148 A206701 A176354

Adjacent sequences:  A348379 A348380 A348381 * A348383 A348384 A348385

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 05 2021

STATUS

approved

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Last modified January 22 02:46 EST 2022. Contains 350481 sequences. (Running on oeis4.)