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A192921 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

%I A192921 #26 Sep 08 2022 08:45:58
%S A192921 1,2,2,7,16,44,113,298,778,2039,5336,13972,36577,95762,250706,656359,
%T A192921 1718368,4498748,11777873,30834874,80726746,211345367,553309352,
%U A192921 1448582692,3792438721,9928733474,25993761698,68052551623,178163893168,466439127884,1221153490481
%N A192921 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.
%C A192921 The titular polynomial is defined by p(n,x) = x*p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=1+x^2.  For discussions of polynomial reduction, see A192232, A192744, and A192872.
%H A192921 Alois P. Heinz, <a href="/A192921/b192921.txt">Table of n, a(n) for n = 0..1000</a>
%H A192921 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A192921 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A192921 G.f.: (1-2*x)*(1+2*x) / ( (1+x)*(1-3*x+x^2) ). - _R. J. Mathar_, May 08 2014
%F A192921 a(n) = A059929(n-1) + 2*A059929(n-2). - _R. J. Mathar_, May 08 2014
%F A192921 a(n) = F(n-4)*F(n) + F(n-1)*F(n+2), where F(-4)=-3, F(-3)=2, F(-2)=-1, F(-1)=1. - _Bruno Berselli_, Nov 03 2015
%F A192921 a(n) = (2^(-n)*(-3*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5. - _Colin Barker_, Oct 01 2016
%e A192921 The coefficients in the polynomials p(n,x) are Fibonacci numbers.  The first seven and their reductions:
%e A192921 ...
%e A192921 1 -> 1
%e A192921 1 + x^2 -> 2 + x
%e A192921 x + x^2 + x^3 -> 2 + 4*x
%e A192921 2*x^2 + x^3 + 2*x^4 -> 7 + 10*x
%e A192921 3*x^3 + 2*x^4 + 3*x^5 -> 16 + 27*x
%e A192921 5*x^4 + 3*x^5 + 5*x^6 -> 44 + 70*x
%e A192921 8*x^5 + 5*x^6 + 8*x^7 -> 113 + 184*x,
%e A192921 so that A192921=(1,2,2,7,16,44,113,...).
%p A192921 a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,2,2>>)[1,1]:
%p A192921 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 28 2016
%t A192921 q = x^2; s = x + 1; z = 28;
%t A192921 p[0, x_] := 1; p[1, x_] := x^2 + 1;
%t A192921 p[n_, x_] := p[n - 1, x]*x + p[n - 2, x]*x^2;
%t A192921 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192921 reduce[{p1_, q_, s_, x_}] :=
%t A192921 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192921        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192921 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192921 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192921   (* A192921 *)
%t A192921 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192921   (* A192879 *)
%t A192921 LinearRecurrence[{2,2,-1}, {1,2,2}, 30] (* _G. C. Greubel_, Feb 06 2019 *)
%o A192921 (PARI) a(n) = round((2^(-n)*(-3*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5) \\ _Colin Barker_, Oct 01 2016
%o A192921 (PARI) Vec(-(2*x-1)*(1+2*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ _Colin Barker_, Oct 01 2016
%o A192921 (PARI) {a(n) = fibonacci(n-2)^2 +fibonacci(n)*fibonacci(n+1)}; \\ _G. C. Greubel_, Feb 06 2019
%o A192921 (Magma) [Fibonacci(n-2)^2 + Fibonacci(n)*Fibonacci(n+1): n in [0..30]]; // _G. C. Greubel_, Feb 06 2019
%o A192921 (Sage) [fibonacci(n-2)^2 +fibonacci(n)*fibonacci(n+1) for n in range(30)] # _G. C. Greubel_, Feb 06 2019
%o A192921 (GAP) List([0..30], n -> Fibonacci(n-2)^2 +Fibonacci(n)*Fibonacci(n+1)); # _G. C. Greubel_, Feb 06 2019
%Y A192921 Cf. A192232, A192744, A192872.
%K A192921 nonn,easy
%O A192921 0,2
%A A192921 _Clark Kimberling_, Jul 12 2011