A192424 a(n) = A192423(n)/2.

1, 0, 2, 1, 8, 10, 39, 70, 208, 439, 1162, 2640, 6641, 15600, 38362, 91481, 222688, 534650, 1295559, 3119990, 7544888, 18194639, 43958642, 106072320, 256167361, 618303360, 1492941842, 3603915601, 8701212248, 21005629450

Offset: 0

Author

Clark Kimberling, Jun 30 2011

Keywords

• nonn
• easy

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 A192423(n) = 2*a(n) = (2, 0, 4, 2, 16, ...) and A192425 = (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, A192423, A192425.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-2*x)/((1+x-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Jul 12 2023

Mathematica

(See A192423.)
LinearRecurrence[{1,4,-1,-1}, {1,0,2,1}, 40] (* G. C. Greubel, Jul 12 2023 *)

SageMath

@CachedFunction
def a(n): # a = A192424
    if (n<4): return (1,0,2,1)[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 12 2023

Formula

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

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

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

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