A192919 Constant term in the reduction by (x^2 -> x+1) of the polynomial F(n+4)*x^n, where F=A000045 (Fibonacci sequence).

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

Offset

0,1

Comments

See A192872.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Formula

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) + a(n+1) = A001906(n+1). - 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

From Amiram Eldar, Oct 06 2020: (Start)

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)

Maple

with(combinat):seq(fibonacci(n-1)*fibonacci(n+4), n=0..27);

Mathematica

(* 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 *)

Prog

(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

Crossrefs

Cf. A000045, A001622, A192232, A192744, A192872, A192883, A192920, A290565.

Sequence in context: A373308 A095123 A019691 * A068607 A303633 A167004

Adjacent sequences: A192916 A192917 A192918 * A192920 A192921 A192922

Keyword

nonn,easy

Author

Clark Kimberling, Jul 12 2011

Status

approved