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Number of compositions (ordered partitions) of n into squarefree parts (A005117) such that no two adjacent parts are equal (Carlitz compositions).
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%I #5 Mar 22 2018 17:56:55

%S 1,1,1,3,3,5,11,15,25,45,69,115,193,309,513,849,1387,2291,3771,6189,

%T 10195,16773,27579,45391,74675,122837,202111,332507,547011,899949,

%U 1480583,2435803,4007361,6592863,10846405,17844319,29357197,48297813,79458705,130724101,215064673

%N Number of compositions (ordered partitions) of n into squarefree parts (A005117) such that no two adjacent parts are equal (Carlitz compositions).

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F G.f.: 1/(1 - Sum_{k>=1} mu(k)^2*x^k/(1 + x^k)), where mu() is the Moebius function (A008683).

%e a(5) = 5 because we have [5], [3, 2], [2, 3], [2, 1, 2] and [1, 3, 1].

%t nmax = 40; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k]^2 x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

%Y Cf. A003242, A005117, A008683, A280194, A301428, A301501.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Mar 22 2018