A192880 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.

1, 0, 3, 7, 34, 123, 495, 1912, 7501, 29253, 114342, 446545, 1744489, 6814224, 26618619, 103979239, 406172770, 1586623227, 6197795703, 24210320296, 94572284197, 369425778645, 1443080391558, 5637075481729, 22019992977457

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

a(n) = Fibonacci(n-1)*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}, {1, 0, 3, 7}, 30] (* G. C. Greubel, Jan 08 2019 *)
Table[Fibonacci[n-1]*LucasL[n, 2]/2, {n, 0, 30}] (* G. C. Greubel, Jul 29 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1+x)*(1-3*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); Coefficients(R!( (1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4) )); // G. C. Greubel, Jan 08 2019
(SageMath) ((1+x)*(1-3*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
(SageMath) [fibonacci(n-1)*lucas_number2(n, 2, -1)/2 for n in (0..30)] # G. C. Greubel, Jul 29 2019
(GAP) a:=[1, 0, 3, 7];; 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, A192882.

Sequence in context: A358961 A024496 A081890 * A355156 A365140 A318444

Adjacent sequences: A192877 A192878 A192879 * A192881 A192882 A192883

Keyword

nonn

Author

Clark Kimberling, Jul 11 2011

Status

approved