A192921 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

1, 2, 2, 7, 16, 44, 113, 298, 778, 2039, 5336, 13972, 36577, 95762, 250706, 656359, 1718368, 4498748, 11777873, 30834874, 80726746, 211345367, 553309352, 1448582692, 3792438721, 9928733474, 25993761698, 68052551623, 178163893168, 466439127884, 1221153490481

Offset: 0

Author

Clark Kimberling, Jul 12 2011

Keywords

• nonn
• easy

Comments

The titular polynomial is defined by p(n,x) = x*p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=1+x^2. For discussions of polynomial reduction, see A192232, A192744, and A192872.

Examples

			The coefficients in the polynomials p(n,x) are Fibonacci numbers.  The first seven and their reductions:
...
1 -> 1
1 + x^2 -> 2 + x
x + x^2 + x^3 -> 2 + 4*x
2*x^2 + x^3 + 2*x^4 -> 7 + 10*x
3*x^3 + 2*x^4 + 3*x^5 -> 16 + 27*x
5*x^4 + 3*x^5 + 5*x^6 -> 44 + 70*x
8*x^5 + 5*x^6 + 8*x^7 -> 113 + 184*x,
so that A192921=(1,2,2,7,16,44,113,...).
		

Links

• Alois P. Heinz, 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

List([0..30], n -> Fibonacci(n-2)^2 +Fibonacci(n)*Fibonacci(n+1)); # G. C. Greubel, Feb 06 2019

Magma

[Fibonacci(n-2)^2 + Fibonacci(n)*Fibonacci(n+1): n in [0..30]]; // G. C. Greubel, Feb 06 2019

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,2,2>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016

Mathematica

q = x^2; s = x + 1; z = 28;
p[0, x_] := 1; p[1, x_] := x^2 + 1;
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}]
  (* A192921 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192879 *)
LinearRecurrence[{2,2,-1}, {1,2,2}, 30] (* G. C. Greubel, Feb 06 2019 *)

PARI

a(n) = round((2^(-n)*(-3*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5) \\ Colin Barker, Oct 01 2016

PARI

Vec(-(2*x-1)*(1+2*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016

PARI

{a(n) = fibonacci(n-2)^2 +fibonacci(n)*fibonacci(n+1)}; \\ G. C. Greubel, Feb 06 2019

Sage

[fibonacci(n-2)^2 +fibonacci(n)*fibonacci(n+1) for n in range(30)] # G. C. Greubel, Feb 06 2019

Formula

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

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

a(n) = A059929(n-1) + 2*A059929(n-2). - R. J. Mathar, May 08 2014

a(n) = F(n-4)*F(n) + F(n-1)*F(n+2), where F(-4)=-3, F(-3)=2, F(-2)=-1, F(-1)=1. - Bruno Berselli, Nov 03 2015

a(n) = (2^(-n)*(-3*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5. - Colin Barker, Oct 01 2016