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A192919
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Constant term in the reduction by (x^2 -> x+1) of the polynomial F(n+4)*x^n, where F=A000045 (Fibonacci sequence).
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2
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3, 0, 8, 13, 42, 102, 275, 712, 1872, 4893, 12818, 33550, 87843, 229968, 602072, 1576237, 4126650, 10803702, 28284467, 74049688, 193864608, 507544125, 1328767778, 3478759198, 9107509827, 23843770272, 62423801000, 163427632717, 427859097162, 1120149658758
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OFFSET
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0,1
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COMMENTS
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = Fibonacci(n-1)*Fibonacci(n+4). - Gary Detlefs, Oct 19 2011
G.f.: (3 -6*x +2*x^2)/((1+x)*(1-3*x+x^2)). - R. J. Mathar, May 08 2014
a(n) = (2^(-n)*(11*(-2)^n-(3-sqrt(5))^n*(-2+sqrt(5))+(2+sqrt(5))*(3+sqrt(5))^n))/5. - Colin Barker, Oct 01 2016
Sum_{n>=2} 1/a(n) = (1/5) * A290565 - 17/150.
Sum_{n>=2} (-1)^n/a(n) = 1/phi - 83/150, where phi is the golden ratio (A001622). (End)
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MAPLE
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with(combinat):seq(fibonacci(n-1)*fibonacci(n+4), n=0..27);
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MATHEMATICA
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(* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 3; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192919 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192920 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n-1]*F[n+4], {n, 0, 30}]] (* G. C. Greubel, Jul 28 2019 *)
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PROG
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(PARI) a(n) = round((2^(-n)*(11*(-2)^n-(3-sqrt(5))^n*(-2+sqrt(5))+(2+sqrt(5))*(3+sqrt(5))^n))/5) \\ Colin Barker, Oct 01 2016
(PARI) Vec((3+2*x^2-6*x)/((1+x)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Oct 01 2016
(PARI) vector(30, n, n--; f=fibonacci; f(n-1)*f(n+4)) \\ G. C. Greubel, Jul 28 2019
(Magma) F:=Fibonacci; [F(n-1)*F(n+4): n in [0..30]]; // G. C. Greubel, Jul 28 2019
(Sage) f=fibonacci; [f(n-1)*f(n+4) for n in (0..30)] # G. C. Greubel, Jul 28 2019
(GAP) F:=Fibonacci;; List([0..30], n-> F(n-1)*F(n+4)); # G. C. Greubel, Jul 28 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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