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Number of compositions of n with no part divisible by 3 and an even number of parts congruent to 4 or 5 modulo 6.
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%I #29 Sep 07 2019 01:35:58

%S 1,1,2,3,5,8,13,22,38,67,120,217,395,722,1323,2428,4460,8197,15070,

%T 27711,50961,93724,172377,317042,583122,1072519,1972660,3628277,

%U 6673431,12274342,22576023,41523768,76374104,140473865,258371706,475219643,874065181,1607656496

%N Number of compositions of n with no part divisible by 3 and an even number of parts congruent to 4 or 5 modulo 6.

%H L. Moser and E. L. Whitney, <a href="https://doi.org/10.4153/CMB-1961-006-0">Weighted compositions</a>, Canad. Math. Bull. 4 (1961), 39-43.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,-1,1)

%F a(n) = (A001590(n+2) + n)/2, see Moser & Whitley reference, Theorem 3.

%F a(n) = A062544(n-3) + n for n >= 3 (also for n = 1 and 2 with A062544(-2) = A062544(-1) = 0), Moser & Whitney.

%F G.f.: (x^5-x^4+x^3-x^2+2*x-1)/((x^3+x^2+x-1)*(x-1)^2). - _Alois P. Heinz_, Sep 06 2019

%e a(4) counts (1,1,1,1), (1,1,2), (1,2,1), (2,1,1), (2,2), but not (1,3) or (3,1) since they contain 3, neither (4) since that has an odd number of parts congruent to 4 or 5 mod 6.

%Y Cf. A001590, A062544.

%K nonn,easy

%O 0,3

%A _Brian Hopkins_, Sep 06 2019