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+2, see A192232.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-1,-1).
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.
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 -> 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, ...).
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 *)
PROG
(Magma) R<x>:=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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 30 2011
STATUS
approved