A192421 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

2, 0, 3, 1, 8, 8, 28, 43, 111, 204, 466, 924, 2007, 4109, 8740, 18136, 38240, 79799, 167643, 350664, 735554, 1540104, 3228459, 6762553, 14172272, 29691368, 62217172, 130356451, 273144327, 572305140, 1199164498, 2512579140, 5264623167, 11030890949

Offset

0,1

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+1, see A192232.

Links

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

Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).

Formula

From Colin Barker, May 12 2014: (Start)

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

G.f.: (2-2*x-3*x^2)/(1-x-3*x^2+x^3+x^4). (End)

a(n) = Sum_{j=0..n} T(n, j)*Fibonacci(j-1), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n. - G. C. Greubel, Jul 11 2023

Example

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 -> 3 + x.
  p(3,x) = 3*x + x^3 -> 1 + 5*x.
  p(4,x) = 2 + 4*x^2 + x^4 -> 8 + 7*x.
From these, read a(n) = (2, 0, 3, 1, 8, ...) and A192422 = (0, 1, 1, 5, 7, ...).

Mathematica

q[x_]:= x+1; d= Sqrt[x^2+4];
p[n_, x_]:= ((x+d)/2)^n + ((x-d)/2)^n (* A162514 *)
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}] (* A192421 *)
Table[Coefficient[Part[t, n], x, 1], {n, 30}] (* A192422 *)
LinearRecurrence[{1, 3, -1, -1}, {2, 0, 3, 1}, 40] (* G. C. Greubel, Jul 11 2023 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-2*x-3*x^2)/(1-x-3*x^2+x^3+x^4) )); // G. C. Greubel, Jul 11 2023
(SageMath)
@CachedFunction
def a(n): # a = A192421
    if (n<4): return (2, 0, 3, 1)[n]
    else: return a(n-1) +3*a(n-2) -a(n-3) -a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 11 2023

Crossrefs

Cf. A162514, A192232, A192422.

Sequence in context: A241640 A158449 A106533 * A035223 A035184 A257541

Adjacent sequences: A192418 A192419 A192420 * A192422 A192423 A192424

Keyword

nonn

Author

Clark Kimberling, Jun 30 2011

Status

approved