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A192883 Constant term in the reduction by (x^2 -> x + 1) of the polynomial F(n+3)*x^n, where F = A000045 (Fibonacci sequence).

%I A192883 #49 Mar 19 2022 03:23:56
%S A192883 2,0,5,8,26,63,170,440,1157,3024,7922,20735,54290,142128,372101,
%T A192883 974168,2550410,6677055,17480762,45765224,119814917,313679520,
%U A192883 821223650,2149991423,5628750626,14736260448,38580030725,101003831720,264431464442,692290561599
%N A192883 Constant term in the reduction by (x^2 -> x + 1) of the polynomial F(n+3)*x^n, where F = A000045 (Fibonacci sequence).
%C A192883 See A192872.
%C A192883 a(n) is also the area of the triangle with vertices at (F(n),F(n+1)), (F(n+1),F(n)), and (F(n+3),F(n+4)) where F(n) = A000045(n). - _J. M. Bergot_, May 22 2014
%H A192883 Colin Barker, <a href="/A192883/b192883.txt">Table of n, a(n) for n = 0..1000</a>
%H A192883 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A192883 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F A192883 a(n) = Fibonacci(n-1) * Fibonacci(n+3). - _Gary Detlefs_, Oct 19 2011
%F A192883 a(n) = Fibonacci(n+1)^2 + (-1)^n. - _Gary Detlefs_, Oct 19 2011
%F A192883 G.f.: ( 2-4*x+x^2 ) / ( (1+x)*(1-3*x+x^2) ). - _R. J. Mathar_, May 07 2014
%F A192883 a(n) = (2^(-1-n)*(7*(-1)^n*2^(1+n) + (3-sqrt(5))^(1+n) + (3+sqrt(5))^(1+n)))/5. - _Colin Barker_, Sep 29 2016
%F A192883 From _Amiram Eldar_, Oct 06 2020: (Start)
%F A192883 Sum_{n>=2} 1/a(n) = 7/18.
%F A192883 Sum_{n>=2} (-1)^n/a(n) = (4/phi - 13/6)/3, where phi is the golden ratio (A001622). (End)
%F A192883 a(n) = a(-2-n) for all n in Z. - _Michael Somos_, Mar 18 2022
%e A192883 G.f. = 2 + 5*x^2 + 8*x^3 + 26*x^4 + 63*x^5 + 170*x^6 + ... - _Michael Somos_, Mar 18 2022
%p A192883 with(combinat):seq(fibonacci(n-1)*fibonacci(n+3), n=0..27): # _Gary Detlefs_, Oct 19 2011
%t A192883 q = x^2; s = x + 1; z = 28;
%t A192883 p[0, x_] := 2; p[1, x_] := 3 x;
%t A192883 p[n_, x_] := p[n - 1, x]*x + p[n - 2, x]*x^2;
%t A192883 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192883 reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192883 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192883 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192883 *)
%t A192883 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* minus A121646 *)
%t A192883 LinearRecurrence[{2,2,-1}, {2,0,5}, 30] (* _G. C. Greubel_, Jan 09 2019 *)
%t A192883 a[ n_] := Fibonacci[n+1]^2 + (-1)^n; (* _Michael Somos_, Mar 18 2022 *)
%o A192883 (PARI) a(n) = round((2^(-1-n)*(7*(-1)^n*2^(1+n)+(3-sqrt(5))^(1+n)+(3+sqrt(5))^(1+n)))/5) \\ _Colin Barker_, Sep 29 2016
%o A192883 (PARI) Vec((2+x^2-4*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ _Colin Barker_, Sep 29 2016
%o A192883 (PARI) {a(n) = fibonacci(n+1)^2 + (-1)^n}; /* _Michael Somos_, Mar 18 2022 */
%o A192883 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( ( 2-4*x+x^2)/((1+x)*(1-3*x+x^2)) )); // _G. C. Greubel_, Jan 09 2019
%o A192883 (Sage) ((2-4*x+x^2 )/((1+x)*(1-3*x+x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 09 2019
%o A192883 (GAP) a:=[2,0,5];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # _G. C. Greubel_, Jan 09 2019
%Y A192883 Cf. A001622, A192232, A192744, A192872.
%K A192883 nonn,easy
%O A192883 0,1
%A A192883 _Clark Kimberling_, Jul 12 2011