1,3

For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

Direct sums can be obtained for A192355 and A192356 in the following way. The polynomials p_{n}(x) can be given in series form by p_{n}(x) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*4*k*x^(n-2*k). For the reduction x^2 -> x+2 then the general form can be seen as x^n -> J_{n}*x + phi_{n}, where J_{n} = A001045(n) are the Jacobsthal numbers and phi_{n} = A078008. The reduction of p_{n}(x) now takes the form p_{n}(x) = x * Sum_{k=0..floor(n/2)} binomial(n,2*k)*4^k*J_{n-2*k} + Sum_{k=0..floor(n/2)} binomial(n,2*k)*4^k*phi_{n-2*k}. Evaluating the series leads to p_{n}(x) = x * (4^n - (-3)^n - 1 + 2^n*delta(n,0))/6 + (4^n + 2*(-3)^n + 2 + 2^n*delta(n,0))/6, where delta(n,k) is the Kronecker delta. - G. C. Greubel, Oct 29 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,11,-12).

Empirical G.f.: x*(2*x^3-5*x^2-2*x+1)/((x-1)*(3*x+1)*(4*x-1)). - Colin Barker, Sep 12 2012

From G. C. Greubel, Oct 28 2018: (Start)

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * 4^k * phi_{n-2*k}, where phi_{n} = A078008(n).a(n) = (4^n + 2*(-3)^n + 2 + 2^n*delta(n,0))/6, with delta(n,0) = 1 if n=0, 0 else. (End)

(See A192352 for a related example.)

q[x_] := x + 2; d = 2;

p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)

Table[Expand[p[n, x]], {n, 0, 6}]

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, 30}]

Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]

(* A192355 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]

(* A192356 *)

Join[{1}, Table[(4^n + 2*(-3)^n + 2)/6, {n, 1, 50}]] (* G. C. Greubel, Oct 20 2018 *)

(PARI) for(n=0, 50, print1(if(n==0, 1, (4^n + 2*(-3)^n + 2)/6), ", ")) \\ G. C. Greubel, Oct 20 2018

(Magma) [1] cat [(4^n + 2*(-3)^n + 2)/6: n in [1..50]]; // G. C. Greubel, Oct 20 2018

Cf. A192232, A192356.

Sequence in context: A242529 A263088 A266231 * A100251 A266607 A290051

Adjacent sequences: A192352 A192353 A192354 * A192356 A192357 A192358

Clark Kimberling, Jun 29 2011

approved