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%I #10 Nov 13 2021 10:22:36
%S 0,0,0,1,3,9,17,41,88,185,387,810,1669,3435,7039,14360,29225,59347,
%T 120228,243166,491085,990446,1995409,4016259,8076959,16231746,
%U 32599773,65437945,131293191,263316897,527912139,1058061751,2120039884,4246934012,8505864639
%N Number of compositions of n that are not a twin (x,x) but have adjacent equal parts.
%C A composition with no adjacent equal parts is also called a Carlitz composition, so these are non-twin, non-Carlitz compositions.
%H A. Knopfmacher and H. Prodinger, <a href="https://core.ac.uk/download/pdf/81957062.pdf">On Carlitz Compositions</a>, Europ. J. Combinatorics (1998) 19, 579-589.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Composition_(combinatorics)">Composition (combinatorics)</a>
%F For n > 0, a(n) = A261983(n) - A059841(n).
%F O.g.f.: 1 + x/(1-2x) - x^2/(1-x^2) - 1/(1 - Sum_{k>0} x^k/(1+x^k)).
%e The a(3) = 1 through a(6) = 17 compositions:
%e (111) (112) (113) (114)
%e (211) (122) (222)
%e (1111) (221) (411)
%e (311) (1113)
%e (1112) (1122)
%e (1121) (1131)
%e (1211) (1221)
%e (2111) (1311)
%e (11111) (2112)
%e (2211)
%e (3111)
%e (11112)
%e (11121)
%e (11211)
%e (12111)
%e (21111)
%e (111111)
%t 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]
%Y Allowing twins gives A261983, complement A003242.
%Y The non-alternating case is A348377, difference A345195.
%Y These compositions are ranked by A348612 \ A007582.
%Y A001250 counts alternating permutations, complement A348615.
%Y A007582 ranks twin compositions.
%Y A011782 counts compositions, strict A032020.
%Y A025047 counts alternating or wiggly compositions, complement A345192.
%Y A051049 counts non-twin compositions, complement A000035(n+1).
%Y A325534 counts separable partitions, ranked by A335433.
%Y A325535 counts inseparable partitions, ranked by A335448.
%Y Cf. A000070, A005649, A059841, A106356, A238279, A333755, A344604, A344614, A344740, A348381.
%K nonn
%O 0,5
%A _Gus Wiseman_, Nov 05 2021
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