A192914 Constant term in the reduction by (x^2 -> x + 1) of the polynomial C(n)*x^n, where C=A000285.

1, 0, 5, 9, 28, 69, 185, 480, 1261, 3297, 8636, 22605, 59185, 154944, 405653, 1062009, 2780380, 7279125, 19057001, 49891872, 130618621, 341963985, 895273340, 2343856029, 6136294753, 16065028224, 42058789925, 110111341545, 288275234716, 754714362597

Offset: 0

Author

Clark Kimberling, Jul 12 2011

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A192232, A192744, A192872.

Programs

GAP

F:=Fibonacci; List([0..30], n -> F(n+1)^2 +F(n)*F(n-3)); # G. C. Greubel, Jan 12 2019

Magma

F:=Fibonacci; [F(n+1)^2+F(n)*F(n-3): n in [0..30]]; // Bruno Berselli, Feb 15 2017

Mathematica

q = x^2; s = x + 1; z = 28;
p[0, x_]:= 1; p[1, x_]:= 4 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}] (* A192914 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* see A192878 *)
LinearRecurrence[{2,2,-1}, {1,0,5}, 30] (* or *) With[{F:= Fibonacci}, Table[F[n+1]^2 +F[n]*F[n-3], {n, 0, 30}]] (* G. C. Greubel, Jan 12 2019 *)

PARI

a(n) = round((2^(-1-n)*(3*(-1)^n*2^(2+n)+(3+sqrt(5))^n*(-1+3*sqrt(5))-(3-sqrt(5))^n*(1+3*sqrt(5))))/5) \\ Colin Barker, Sep 29 2016

PARI

Vec((1+3*x^2-2*x)/((1+x)*(x^2-3*x+1)) + O(x^30)) \\ Colin Barker, Sep 29 2016

PARI

{f=fibonacci}; vector(30, n, n--; f(n+1)^2 +f(n)*f(n-3)) \\ G. C. Greubel, Jan 12 2019

Sage

f=fibonacci; [f(n+1)^2 +f(n)*f(n-3) for n in (0..30)] # G. C. Greubel, Jan 12 2019

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).

G.f.: (1 + 3*x^2 - 2*x)/((1 + x)*(x^2 - 3*x + 1)). - R. J. Mathar, May 08 2014

a(n) = (2^(-1-n)*(3*(-1)^n*2^(2+n) + (3 + sqrt(5))^n*(-1 + 3*sqrt(5)) - (3-sqrt(5))^n*(1 + 3*sqrt(5))))/5. - Colin Barker, Sep 29 2016

a(n) = F(n+1)^2 + F(n)*F(n-3). - Bruno Berselli, Feb 15 2017