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A154691 Expansion of (1+x+x^2) / ((1-x)*(1-x-x^2)).

%I A154691 #38 Apr 09 2025 14:14:33
%S A154691 1,3,7,13,23,39,65,107,175,285,463,751,1217,1971,3191,5165,8359,13527,
%T A154691 21889,35419,57311,92733,150047,242783,392833,635619,1028455,1664077,
%U A154691 2692535,4356615,7049153,11405771,18454927,29860701,48315631,78176335
%N A154691 Expansion of (1+x+x^2) / ((1-x)*(1-x-x^2)).
%H A154691 Vincenzo Librandi, <a href="/A154691/b154691.txt">Table of n, a(n) for n = 0..1000</a>
%H A154691 Tomislav Došlić and Biserka Kolarec, <a href="https://doi.org/10.3390/math13071179">On Log-Definite Tempered Combinatorial Sequences</a>, Mathematics (2025) Vol. 13, Iss. 7, 1179.
%H A154691 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F A154691 a(n+1) - a(n) = A006355(n+3) = A055389(n+3).
%F A154691 a(n) = A066629(n-1) + A066629(n).
%F A154691 a(n) = A006355(n+4) - 3 = A078642(n+1) - 3.
%F A154691 a(n+1) = a(n) + 2*A000045(n+2). - _Reinhard Zumkeller_, Nov 17 2013
%F A154691 From _Colin Barker_, Feb 01 2017: (Start)
%F A154691 a(n) = -3 + (2^(1-n)*((1-r)^n*(-2+r) + (1+r)^n*(2+r))) / r where r=sqrt(5).
%F A154691 a(n) = 2*a(n-1) - a(n-3) for n>2. (End)
%F A154691 a(n) = 2*Fibonacci(n+3) - 3. - _Greg Dresden_, Oct 10 2020
%F A154691 E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 3*exp(x). - _Stefano Spezia_, Apr 09 2025
%p A154691 A154691 := proc(n) coeftayl( (1+x+x^2)/(1-x-x^2)/(1-x),x=0,n) ; end proc:
%t A154691 Fibonacci[Range[3,60]]*2 -3 (* _Vladimir Joseph Stephan Orlovsky_, Mar 19 2010 *)
%t A154691 CoefficientList[Series[(1 + x + x^2)/((1 - x - x^2)(1 - x)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 18 2012 *)
%o A154691 (Haskell)
%o A154691 a154691 n = a154691_list !! n
%o A154691 a154691_list = 1 : zipWith (+)
%o A154691                    a154691_list (drop 2 $ map (* 2) a000045_list)
%o A154691 -- _Reinhard Zumkeller_, Nov 17 2013
%o A154691 (PARI) Vec((1+x+x^2) / ((1-x-x^2)*(1-x)) + O(x^60)) \\ _Colin Barker_, Feb 01 2017
%o A154691 (Magma)
%o A154691 A154691:= func< n | 2*Fibonacci(n+3) - 3 >;
%o A154691 [A154691(n): n in [0..40]]; // _G. C. Greubel_, Jan 18 2025
%o A154691 (Python)
%o A154691 def A154691(n): return 2*fibonacci(n+3) - 3
%o A154691 print([A154691(n) for n in range(41)]) # _G. C. Greubel_, Jan 18 2025
%Y A154691 Cf. A000045, A001595, A006355, A055389, A066629, A078642, A166863.
%K A154691 easy,nonn
%O A154691 0,2
%A A154691 _R. J. Mathar_, Jan 14 2009