OFFSET
1,3
COMMENTS
The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+5). 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 = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,12,-8,-16).
FORMULA
From Colin Barker, May 11 2014: (Start)
a(n) = 2*a(n-1) + 12*a(n-2) - 8*a(n-3) - 16*a(n-4).
G.f.: x*(1-2*x-4*x^2)/(1-2*x-12*x^2+8*x^3+16*x^4). (End)
From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+5))^n - (x-sqrt(x+5))^n)/(2*sqrt(x+5)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1). (End)
EXAMPLE
The first five polynomials p(n,x) and their reductions are as follows:
p(0, x) = 1 -> 1
p(1, x) = 2*x -> 2*x
p(2, x) = 3 + x + 3*x^2 -> 8 + 4*x
p(3, x) = 12*x + 4*x^2 + 4*x^3 -> 8 + 32*x
p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 96 + 96*x.
MATHEMATICA
q[x_]:= x+1; d= Sqrt[x+5];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n, 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, 1, 30}];
Table[Coefficient[Part[t, n], x, 0], {n, 30}] (* A192386 *)
Table[Coefficient[Part[t, n], x, 1], {n, 30}] (* A192387 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 30}] (* A192388 *)
LinearRecurrence[{2, 12, -8, -16}, {1, 0, 8, 8}, 40] (* G. C. Greubel, Jul 10 2023 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Integers(), 41);
Coefficients(R!( x*(1-2*x-4*x^2)/(1-2*x-12*x^2+8*x^3+16*x^4) )); // G. C. Greubel, Jul 10 2023
(SageMath)
@CachedFunction
def a(n): # a = A192386
if (n<5): return (0, 1, 0, 8, 8)[n]
else: return 2*a(n-1) +12*a(n-2) -8*a(n-3) -16*a(n-4)
[a(n) for n in range(1, 41)] # G. C. Greubel, Jul 10 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 30 2011
STATUS
approved