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A192879 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

%I A192879 #43 Oct 29 2023 16:56:14
%S A192879 0,1,4,10,27,70,184,481,1260,3298,8635,22606,59184,154945,405652,
%T A192879 1062010,2780379,7279126,19057000,49891873,130618620,341963986,
%U A192879 895273339,2343856030,6136294752,16065028225,42058789924,110111341546,288275234715,754714362598
%N A192879 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.
%C A192879 The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x+1, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1.  See A192872.
%C A192879 A192879 is also generated as the coefficient sequence of x in the reduction x^2->x+1 of the polynomial v(n,x) defined by v(0,x) = 2, v(1,x) = x+1, and v(n,x) = x*v(n-1,x) + 2*(x^2)*v(n-1,x) + 1, for n>0, v(n,x) = F(n)*x^(n-1) + L(n)*x^n, where F(n) = A000045(n) (Fibonacci numbers) and L(n) = A000032(n) (Lucas numbers).
%H A192879 Colin Barker, <a href="/A192879/b192879.txt">Table of n, a(n) for n = 0..1000</a>
%H A192879 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A192879 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), with a(0) = 0, a(1) = 1, a(2) = 4.
%F A192879 G.f.: x * (1+2*x) / ((1+x) * (1-3*x+x^2)). - _Colin Barker_, Jun 18 2012
%F A192879 a(n) = (2^(-1-n) * ((-1)^n*2^(1+n) + (3+sqrt(5))^n * (-1+3*sqrt(5)) - (3-sqrt(5))^n * (1+3*sqrt(5))))/5. - _Colin Barker_, Sep 29 2016
%F A192879 a(n) = F(n-1)*F(n) + F(2n), where F(n) is a Fibonacci number. - _Rigoberto Florez_, Feb 06 2020
%F A192879 E.g.f.: (exp(-x) + exp(3*x/2) * (3*sqrt(5)*sinh(sqrt(5)*x/2) - cosh(sqrt(5)*x/2)))/5. - _Stefano Spezia_, Feb 06 2020
%F A192879 a(n)*F(n) = the number of ways to tile a 3-arm starfish (with n-1 cells on each arm and one cell in the center) using squares and dominos. - _Greg Dresden_ and Hasita Kanamarlapudi, Oct 02 2023
%e A192879 The first six polynomials and reductions:
%e A192879   p(0,x) = 3 -> 3
%e A192879   p(1,x) = x -> x
%e A192879   p(2,x) = 4*x^2 -> 4+4*x
%e A192879   p(3,x) = 5*x^3 -> 5+10*x
%e A192879   p(4,x) = 9*x^4 -> 18+27*x
%e A192879   p(5,x) = 14*x^5 -> 42+27*x
%e A192879 In general, p(n,x) = (A104449(n))*x^n -> A192878(n) + A192879(n)*x.
%p A192879 with(combinat); seq( fibonacci(2*n) + fibonacci(n)*fibonacci(n-1), n=0..40); # _G. C. Greubel_, Feb 13 2020
%t A192879 (See A192878.)
%t A192879 LinearRecurrence[{2,2,-1}, {0,1,4}, 30] (* _G. C. Greubel_, Jan 07 2019 *)
%t A192879 a[n_] := a[n] = 2*a[n-1]+2*a[n - 2]-a[n-3]; a[0] = 0; a[1]=1; a[2]=4; Table[a[n], {n,0,40}] (* _Rigoberto Florez_, Feb 06 2020 *)
%t A192879 Table[Fibonacci[n]*Fibonacci[n-1]+Fibonacci[2n], {n,0,40}] (* _Rigoberto Florez_, Feb 06 2020 *)
%o A192879 (PARI) a(n) = round((2^(-1-n)*((-1)^n*2^(1+n)+(3+sqrt(5))^n*(-1+3*sqrt(5))-(3-sqrt(5))^n*(1+3*sqrt(5))))/5) \\ _Colin Barker_, Sep 29 2016
%o A192879 (PARI) concat(0, Vec(x*(1+2*x)/((1+x)*(1-3*x+x^2)) + O(x^40))) \\ _Colin Barker_, Sep 29 2016
%o A192879 (Magma) I:=[0,1,4]; [n le 3 select I[n] else 2*Self(n-1) +2*Self(n-2) -Self(n-3): n in [1..40]]; // _G. C. Greubel_, Jan 07 2019
%o A192879 (Sage) (x*(1+2*x)/((1+x)*(1-3*x+x^2))).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 07 2019
%o A192879 (GAP) a:=[0,1,4];; for n in [4..40] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # _G. C. Greubel_, Jan 07 2019
%Y A192879 Cf. A192872, A192878.
%K A192879 nonn,easy
%O A192879 0,3
%A A192879 _Clark Kimberling_, Jul 11 2011