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A116698 Expansion of (1-x+3*x^2+x^3) / ((1-x-x^2)*(1+2*x^2)).

%I A116698 #21 Aug 26 2025 09:53:20
%S A116698 1,0,2,5,5,4,13,29,34,39,89,176,233,313,610,1115,1597,2328,4181,7277,
%T A116698 10946,16687,28657,48416,75025,117297,196418,326003,514229,815656,
%U A116698 1346269,2211077,3524578,5637351,9227465
%N A116698 Expansion of (1-x+3*x^2+x^3) / ((1-x-x^2)*(1+2*x^2)).
%H A116698 Colin Barker, <a href="/A116698/b116698.txt">Table of n, a(n) for n = 0..1000</a>
%H A116698 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,2,2).
%F A116698 a(2*n) = A000045(2*n+1) = A001519(n).
%F A116698 a(n) = a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) for n > 3. - _Colin Barker_, May 18 2019
%F A116698 From _G. C. Greubel_, Aug 24 2025: (Start)
%F A116698 a(n) = A000045(n+1) - (-1)^floor((n-1)/2) * (n mod 2) * 2^floor(n/2).
%F A116698 E.g.f.: exp(x/2)*(cosh(sqrt(5)*x/2) + (1/sqrt(5))*sinh(sqrt(5)*x/2))  - sin(sqrt(2)*x)/sqrt(2). (End)
%t A116698 CoefficientList[Series[(1-x+3x^2+x^3)/((1-x-x^2)(1+2x^2)),{x,0,100}],x] (* or *) LinearRecurrence[{1,-1,2,2},{1,0,2,5},100] (* _Harvey P. Dale_, May 14 2022 *)
%t A116698 Table[Fibonacci[n+1] -I^(n-1)*Mod[n,2]*2^Floor[n/2], {n,0,50}] (* _G. C. Greubel_, Aug 24 2025 *)
%o A116698 (PARI) Vec((1-x +3*x^2 +x^3)/((1-x-x^2)*(1+2*x^2)) + O(x^40)) \\ _Colin Barker_, May 18 2019
%o A116698 (Magma)
%o A116698 A116698:= func< n | Fibonacci(n+1) -((n mod 2) -2*0^((n+1) mod 4))*2^Floor(n/2) >;
%o A116698 [A116898(n): n in [0..50]]; // _G. C. Greubel_, Aug 24 2025
%o A116698 (SageMath)
%o A116698 def A116898(n): return fibonacci(n+1) - (-1)**((n-1)//2)*(n%2)*2**(n//2)
%o A116698 print([A116898(n) for n in range(51)]) # _G. C. Greubel_, Aug 24 2025
%Y A116698 Cf. A000045, A001519, A006498, A115008, A116697, A116699.
%K A116698 easy,nonn,changed
%O A116698 0,3
%A A116698 _Creighton Dement_, Feb 23 2006