A192423 Constant term of the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.

2, 0, 4, 2, 16, 20, 78, 140, 416, 878, 2324, 5280, 13282, 31200, 76724, 182962, 445376, 1069300, 2591118, 6239980, 15089776, 36389278, 87917284, 212144640, 512334722, 1236606720, 2985883684, 7207831202, 17402424496, 42011258900

Offset: 0

Author

Clark Kimberling, Jun 30 2011

Keywords

• nonn
• easy

Comments

The polynomial p(n,x) is defined by ((x+d)/2)^n + ((x-d)/2)^n, where d = sqrt(x^2+4). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+2, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x) = 2 -> 2
  p(1,x) = x -> x
  p(2,x) = 2 + x^2 -> 4 + x
  p(3,x) = 3*x + x^3 -> 2 + 6*x
  p(4,x) = 2 + 4*x^2 + x^4 -> 16 + 9*x.
From these, read a(n) = (2, 0, 4, 2, 16, ...) and A192424 = (0, 1, 1, 6, 9, ...).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-1).

Crossrefs

Cf. A000032, A000129, A078008, A161514, A192232, A192424, A192425.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 2*(1+x)*(1-2*x)/((1+x-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Jul 11 2023

Mathematica

q[x_]:= x+2; d= Sqrt[x^2+4];
p[n_, x_]:= ((x+d)/2)^n + ((x-d)/2)^n (* A161514 *)
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[FixedPoint[Expand[#1/. reductionRules] &, p[n,x]], {n,0,30}]
Table[Coefficient[Part[t, n], x, 0], {n,30}]   (* A192423 *)
Table[Coefficient[Part[t, n], x, 1], {n,30}]   (* A192424 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n,30}]   (* A192425 *)
LinearRecurrence[{1,4,-1,-1}, {2,0,4,2}, 40] (* G. C. Greubel, Jul 11 2023 *)

SageMath

@CachedFunction
def a(n): # a = A192423
    if (n<4): return (2,0,4,2)[n]
    else: return a(n-1) +4*a(n-2) -a(n-3) -a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 11 2023

Formula

From Colin Barker, May 11 2014: (Start)

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

G.f.: 2*(1+x)*(1-2*x) / ((1+x-x^2)*(1-2*x-x^2)). (End)

From G. C. Greubel, Jul 11 2023: (Start)

a(n) = Sum_{j=0..n} T(n, j)*A078008(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n.

a(n) = (2/3)*((-1)^n*A000032(n) + A000129(n+1) - A000129(n)). (End)