A192879 Coefficient of x in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

0, 1, 4, 10, 27, 70, 184, 481, 1260, 3298, 8635, 22606, 59184, 154945, 405652, 1062010, 2780379, 7279126, 19057000, 49891873, 130618620, 341963986, 895273339, 2343856030, 6136294752, 16065028225, 42058789924, 110111341546, 288275234715, 754714362598

Offset: 0

Author

Clark Kimberling, Jul 11 2011

Keywords

• nonn
• easy

Comments

The polynomial p(n,x) is defined by p(0,x) = 1, p(1,x) = x+1, and p(n,x) = x*p(n-1,x) + 2*(x^2)*p(n-1,x) + 1. See A192872.

A192879 is also generated as the coefficient sequence of x in the reduction x^2->x+1 of the polynomial v(n,x) defined by v(0,x) = 2, v(1,x) = x+1, and v(n,x) = x*v(n-1,x) + 2*(x^2)*v(n-1,x) + 1, for n>0, v(n,x) = F(n)*x^(n-1) + L(n)*x^n, where F(n) = A000045(n) (Fibonacci numbers) and L(n) = A000032(n) (Lucas numbers).

Examples

			The first six polynomials and reductions:
  p(0,x) = 3 -> 3
  p(1,x) = x -> x
  p(2,x) = 4*x^2 -> 4+4*x
  p(3,x) = 5*x^3 -> 5+10*x
  p(4,x) = 9*x^4 -> 18+27*x
  p(5,x) = 14*x^5 -> 42+27*x
In general, p(n,x) = (A104449(n))*x^n -> A192878(n) + A192879(n)*x.
		

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. A192872, A192878.

Programs

GAP

a:=[0,1,4];; for n in [4..40] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jan 07 2019

Magma

I:=[0,1,4]; [n le 3 select I[n] else 2*Self(n-1) +2*Self(n-2) -Self(n-3): n in [1..40]]; // G. C. Greubel, Jan 07 2019

Maple

with(combinat); seq( fibonacci(2*n) + fibonacci(n)*fibonacci(n-1), n=0..40); # G. C. Greubel, Feb 13 2020

Mathematica

(See A192878.)
LinearRecurrence[{2,2,-1}, {0,1,4}, 30] (* G. C. Greubel, Jan 07 2019 *)
a[n_] := a[n] = 2*a[n-1]+2*a[n - 2]-a[n-3]; a[0] = 0; a[1]=1; a[2]=4; Table[a[n], {n,0,40}] (* Rigoberto Florez, Feb 06 2020 *)
Table[Fibonacci[n]*Fibonacci[n-1]+Fibonacci[2n], {n,0,40}] (* Rigoberto Florez, Feb 06 2020 *)

PARI

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

PARI

concat(0, Vec(x*(1+2*x)/((1+x)*(1-3*x+x^2)) + O(x^40))) \\ Colin Barker, Sep 29 2016

Sage

(x*(1+2*x)/((1+x)*(1-3*x+x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jan 07 2019

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), with a(0) = 0, a(1) = 1, a(2) = 4.

G.f.: x * (1+2*x) / ((1+x) * (1-3*x+x^2)). - Colin Barker, Jun 18 2012

a(n) = (2^(-1-n) * ((-1)^n*2^(1+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)*F(n) + F(2n), where F(n) is a Fibonacci number. - Rigoberto Florez, Feb 06 2020

E.g.f.: (exp(-x) + exp(3*x/2) * (3*sqrt(5)*sinh(sqrt(5)*x/2) - cosh(sqrt(5)*x/2)))/5. - Stefano Spezia, Feb 06 2020

a(n)*F(n) = the number of ways to tile a 3-arm starfish (with n-1 cells on each arm and one cell in the center) using squares and dominos. - Greg Dresden and Hasita Kanamarlapudi, Oct 02 2023