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A192383
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Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
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3
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1, 0, 6, 8, 60, 160, 744, 2496, 10064, 36480, 140512, 522624, 1983168, 7439360, 28091520, 105674752, 398391552, 1500057600, 5652182528, 21288560640, 80200784896, 302101094400, 1138045495296, 4286942363648, 16149041172480, 60833034895360
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OFFSET
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1,3
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COMMENTS
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The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d=sqrt(x+3). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
G.f.: x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4). (End)
T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1). (End)
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EXAMPLE
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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 -> 6 + 4*x
p(3, x) = 12*x + 4*x^2 + 4*x^3 -> 8 + 24*x
p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x.
From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).
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MATHEMATICA
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q[x_]:= x+1; d= Sqrt[x+3];
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}] (* A192383 *)
Table[Coefficient[Part[t, n], x, 1], {n, 30}] (* A192384 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 30}] (* A192385 *)
LinearRecurrence[{2, 8, -4, -4}, {1, 0, 6, 8}, 40] (* G. C. Greubel, Jul 10 2023 *)
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PROG
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(Magma)
R<x>:=PowerSeriesRing(Integers(), 41);
Coefficients(R!( x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4) )) // G. C. Greubel, Jul 10 2023
(SageMath)
@CachedFunction
if (n<5): return (0, 1, 0, 6, 8)[n]
else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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