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A364705 Expansion of 1/(1 - 4*x - x^2 + x^3).

%I A364705 #13 Mar 27 2025 23:27:36
%S A364705 1,4,17,71,297,1242,5194,21721,90836,379871,1588599,6643431,27782452,
%T A364705 116184640,485877581,2031912512,8497342989,35535406887,148607058025,
%U A364705 621466295998,2598936835130,10868606578493,45451896853104,190077257155779,794892318897727,3324194635893583
%N A364705 Expansion of 1/(1 - 4*x - x^2 + x^3).
%H A364705 G. C. Greubel, <a href="/A364705/b364705.txt">Table of n, a(n) for n = 0..1000</a>
%H A364705 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,1,-1).
%F A364705 G.f.: 1/(1 - 4*x - x^2 + x^3).
%F A364705 a(n) = 4*a(n-1) + a(n-2) - a(n-3).
%F A364705 a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n-k, j)*binomial(n-k, k-j)*4^(n-2*k)*((1-sqrt(17))/2)^(k-j)*((1+sqrt(17))/2)^j.
%t A364705 LinearRecurrence[{4,1,-1}, {1,4,17}, 41]
%o A364705 (Magma) I:=[1,4,17]; [n le 3 select I[n] else 4*Self(n-1) +Self(n-2) -Self(n-3): n in [1..41]];
%o A364705 (SageMath)
%o A364705 @CachedFunction
%o A364705 def a(n): # a = A364705
%o A364705     if (n<3): return (1,4,17)[n]
%o A364705     else: return 4*a(n-1) +a(n-2) -a(n-3)
%o A364705 [a(n) for n in range(41)]
%Y A364705 Cf. A124807, A126393.
%K A364705 nonn
%O A364705 0,2
%A A364705 _G. C. Greubel_, Aug 04 2023