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

0,3

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).

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), 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

Example

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.

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Cf. A192872, A192878.

Sequence in context: A192210 A121494 A122744 * A077923 A183325 A052982

Adjacent sequences: A192876 A192877 A192878 * A192880 A192881 A192882

Keyword

nonn,easy

Author

Clark Kimberling, Jul 11 2011

Status

approved