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A192426
Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
2
2, 0, 5, 1, 18, 13, 81, 106, 413, 729, 2258, 4653, 12833, 28666, 74493, 173545, 437346, 1041421, 2583089, 6221322, 15304541, 37079289, 90826994, 220729069, 539487297, 1313161498, 3205831869, 7809748489, 19054635650, 46439068365
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+8). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.
FORMULA
From Colin Barker, May 12 2014: (Start)
a(n) = a(n-1) + 5*a(n-2) - 2*a(n-3) - 4*a(n-4).
G.f.: (2-2*x-5*x^2)/(1-x-5*x^2+2*x^3+4*x^4). (End)
a(n) = Sum_{k=0..n} T(n, k)*Fibonacci(k-1), where T(n, k) = [x^k] ((x + sqrt(x^2+8))^n + (x - sqrt(x^2+8))^n)/2^n. - G. C. Greubel, Jul 12 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) = 4 + x^2 -> 5 + x
p(3,x) = 6*x + x^3 -> 1 + 8*x
p(4,x) = 8 + 8*x^2 + x^4 -> 18 + 11*x.
From these, read a(n) = (2, 0, 5, 1, 18, ...) and A192427 = (0, 1, 1, 8, 11, ...).
MATHEMATICA
q[x_]:= x+1; d= Sqrt[x^2+8];
p[n_, x_]:= ((x+d)/2)^n + ((x-d)/2)^n (* suggested by 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}] (* A192426 *)
Table[Coefficient[Part[t, n], x, 1], {n, 30}] (* A192427 *)
LinearRecurrence[{1, 5, -2, -4}, {2, 0, 5, 1}, 40] (* G. C. Greubel, Jul 12 2023 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-2*x-5*x^2)/(1-x-5*x^2+2*x^3+4*x^4) )); // G. C. Greubel, Jul 12 2023
(SageMath)
@CachedFunction
def a(n): # a = A192426
if (n<4): return (2, 0, 5, 1)[n]
else: return a(n-1) + 5*a(n-2) - 2*a(n-3) - 4*a(n-4)
[a(n) for n in range(41)] # G. C. Greubel, Jul 12 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 30 2011
EXTENSIONS
Typo in name corrected by G. C. Greubel, Jul 12 2023
STATUS
approved