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

2, 0, 5, 8, 26, 63, 170, 440, 1157, 3024, 7922, 20735, 54290, 142128, 372101, 974168, 2550410, 6677055, 17480762, 45765224, 119814917, 313679520, 821223650, 2149991423, 5628750626, 14736260448, 38580030725, 101003831720, 264431464442, 692290561599

Offset: 0

Author

Clark Kimberling, Jul 12 2011

Keywords

• nonn
• easy

Comments

See A192872.

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

Examples

			G.f. = 2 + 5*x^2 + 8*x^3 + 26*x^4 + 63*x^5 + 170*x^6 + ... - _Michael Somos_, Mar 18 2022
		

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. A001622, A192232, A192744, A192872.

Programs

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

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( ( 2-4*x+x^2)/((1+x)*(1-3*x+x^2)) )); // G. C. Greubel, Jan 09 2019

Maple

with(combinat):seq(fibonacci(n-1)*fibonacci(n+3), n=0..27): # Gary Detlefs, Oct 19 2011

Mathematica

q = x^2; s = x + 1; z = 28;
p[0, x_] := 2; p[1, x_] := 3 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}] (* A192883 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* minus A121646 *)
LinearRecurrence[{2,2,-1}, {2,0,5}, 30] (* G. C. Greubel, Jan 09 2019 *)
a[ n_] := Fibonacci[n+1]^2 + (-1)^n; (* Michael Somos, Mar 18 2022 *)

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

PARI

Vec((2+x^2-4*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Sep 29 2016

PARI

{a(n) = fibonacci(n+1)^2 + (-1)^n}; /* Michael Somos, Mar 18 2022 */

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

Formula

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

a(n) = Fibonacci(n-1) * Fibonacci(n+3). - Gary Detlefs, Oct 19 2011

a(n) = Fibonacci(n+1)^2 + (-1)^n. - Gary Detlefs, Oct 19 2011

G.f.: ( 2-4*x+x^2 ) / ( (1+x)*(1-3*x+x^2) ). - R. J. Mathar, May 07 2014

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

From Amiram Eldar, Oct 06 2020: (Start)

Sum_{n>=2} 1/a(n) = 7/18.

Sum_{n>=2} (-1)^n/a(n) = (4/phi - 13/6)/3, where phi is the golden ratio (A001622). (End)

a(n) = a(-2-n) for all n in Z. - Michael Somos, Mar 18 2022