A052554 -id:A052554 - cp's OEIS frontend

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

Clark Kimberling, Jul 10 2011

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).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-12,8).

Cf. A192744, A192232, A192808 - A192811.

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

m:=30; R:=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

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  (* A192811 *)
LinearRecurrence[{7,-12,8}, {1,2,6}, 50] (* G. C. Greubel, Jan 02 2019 *)

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

((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

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

A192252 0-sequence of reduction of (n!) by x^2 -> x+1.

Original entry on oeis.org

1, 1, 3, 9, 57, 417, 4017, 44337, 568497, 8188977, 131568177, 2326992177, 44958134577, 941649129777, 21254190979377, 514247427715377, 13277149259395377, 364340640790147377, 10588931448837763377, 324919870905259651377, 10496883167091791491377

Offset: 0

Clark Kimberling, Jun 27 2011

nonn

See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".

After the tenth term, the final digit is 7, for terms in both A192252 and A192253. After the 100th term, the final 6 digits of each term of A192252 are 9,3,1,3,7,7.

			The sequence (n!)=(1,1,2,6,24,120,...) provides coefficients for the power series 1+x+2x^2+6x^3+..., of which the (n+1)st partial sum is the polynomial p(x)=1+x+2x^2+...+(n!)x^n, of which reduction by x^2 -> x+1 (as presented at A192232) is A192252(n)+x*A192253(n).

Cf. A192232, A192253.

c[n_] := n!; (* A000142 *)
Table[c[n], {n, 1, 15}]
q[x_] := x + 1;
p[0, x_] := 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 50}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 50}]  (* A192252 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 50}]  (* A192253 *)
Table[Coefficient[(-7 + Part[t, n])/10, x, 0], {n, 1, 30}]
(* Peter J. C. Moses, Jun 20 2011 *)

Conjecture: a(n) +(-n-1)*a(n-1) -n*(n-2)*a(n-2) +n*(n-1)*a(n-3)=0. - R. J. Mathar, May 04 2014

Conjecture: a(n) = Sum_{k=0..n} A052554(k). - Sean A. Irvine, Jul 14 2022

cp's OEIS Frontend

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

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