A192808 Constant term in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2. See Comments.

1, 2, 6, 26, 126, 618, 3022, 14746, 71902, 350538, 1708910, 8331130, 40615294, 198004778, 965298958, 4705957722, 22942154782, 111845982474, 545263681710, 2658231220538, 12959222223038, 63177890368490, 308000415667278, 1501542003033370

Offset

0,2

Comments

For discussions of polynomial reduction, see A192232 and A192744.

If the reduction (x^2 + c)^n by x^3 -> x^2 + c is applied to the polynomials (x^2+c)^n for c=1 instead of c=2, the results are as follows:

A052554: constant terms,

A052529: coefficients of x,

A124820: coefficients of x^2.

Those three sequences satisfy the recurrence:

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

Links

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

Index entries for linear recurrences with constant coefficients, signature (7,-12,8).

Formula

a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).

G.f.: (1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3). - Colin Barker, Jul 26 2012

Mathematica

q = x^3; s = x^2 + 2; z = 40;
p[n_, x_] := (x^2 + 2)^n;
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}] (* A192808 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192809 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192810 *)
uu = u2/2, {n, 1, z}]  (* A192811 *)
LinearRecurrence[{7, -12, 8}, {1, 2, 6}, 50] (* G. C. Greubel, Jan 02 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3)) \\ G. C. Greubel, Jan 02 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3) )); // G. C. Greubel, Jan 02 2019
(Sage) ((1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 02 2019
(GAP) a:=[1, 2, 6];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 02 2019

Crossrefs

Cf. A192744, A192232, A192808 - A192811.

Sequence in context: A208034 A092880 A370185 * A034474 A306041 A345870

Adjacent sequences: A192805 A192806 A192807 * A192809 A192810 A192811

Keyword

nonn,easy

Author

Clark Kimberling, Jul 10 2011

Status

approved