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Expansion of (1-x)^2/((1-x)^3 - 4*x^3).
1

%I #23 May 06 2022 13:53:47

%S 1,1,1,5,17,41,97,253,673,1745,4481,11573,30001,77689,200929,519725,

%T 1344833,3479969,9004033,23296357,60276817,155961545,403535969,

%U 1044107357,2701521889,6989923441,18085741441,46795063445,121077583217

%N Expansion of (1-x)^2/((1-x)^3 - 4*x^3).

%H Seiichi Manyama, <a href="/A097123/b097123.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,5)

%F G.f.: (1-2*x+x^2)/(1-3*x+3*x^2-5*x^3).

%F a(n) = 3*a(n-1) - 3*a(n-2) + 5*a(n-3).

%F a(n) = Sum_{k=0..floor(n/3)} binomial(n, 3k) * 4^k.

%t LinearRecurrence[{3, -3, 5}, {1, 1, 1}, 30] (* _Amiram Eldar_, Oct 11 2021 *)

%t CoefficientList[Series[(1-x)^2/((1-x)^3-4x^3),{x,0,30}],x] (* _Harvey P. Dale_, May 06 2022 *)

%o (PARI) a(n) = sum(k=0, n\3, binomial(n, 3*k) * 4^k); \\ _Michel Marcus_, Oct 11 2021

%Y Cf. A097122.

%K easy,nonn

%O 0,4

%A _Paul Barry_, Jul 25 2004