A192882 Coefficient of x in the reduction by (x^2 -> x+1) of the polynomial p(n,x) given in Comments.

0, 1, 3, 14, 51, 205, 792, 3107, 12117, 47362, 184965, 722591, 2822544, 11025793, 43069611, 168242270, 657200859, 2567211037, 10028243016, 39173122739, 153021167805, 597743469778, 2334953116653, 9120979734623, 35629097057568

Offset

0,3

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

Formula

a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).

G.f.: x*(1+x+x^2) / ( 1-2*x-7*x^2-2*x^3+x^4 ). - R. J. Mathar, May 07 2014

a(n) = Fibonacci(n)*Pell-Lucas(n)/2, where Pell-Lucas(n) = A002203(n). - G. C. Greubel, Jul 29 2019

Mathematica

(* First program *)
q = x^2; s = x + 1; z = 25;
p[0, x_]:= 1; p[1, x_]:= x;
p[n_, x_]:= 2 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}] (* A192880 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192882 *)
FindLinearRecurrence[u1]
FindLinearRecurrence[u2]
(* Additional programs *)
LinearRecurrence[{2, 7, 2, -1}, {0, 1, 3, 14}, 30] (* G. C. Greubel, Jan 08 2019 *)
Table[Fibonacci[n]*LucasL[n, 2]/2, {n, 0, 30}] (* G. C. Greubel, Jul 29 2019 *)

Prog

(PARI) my(x='x+O('x^30)); concat([0], Vec(x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3 +x^4))) \\ G. C. Greubel, Jan 08 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3+x^4) )); // G. C. Greubel, Jan 08 2019
(Sage) (x*(1+x+x^2)/(1-2*x-7*x^2-2*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019
(GAP) a:=[0, 1, 3, 14];; for n in [5..30] do a[n]:=2*a[n-1]+7*a[n-2] +2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jan 08 2019

Crossrefs

Cf. A000045, A002203, A192872, A192880.

Sequence in context: A359253 A320826 A322199 * A351690 A056076 A117133

Adjacent sequences: A192879 A192880 A192881 * A192883 A192884 A192885

Keyword

nonn

Author

Clark Kimberling, Jul 11 2011

Status

approved