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A192425
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Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.
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6
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0, 1, 1, 6, 9, 31, 60, 169, 369, 954, 2201, 5479, 12960, 31721, 75881, 184326, 443169, 1072871, 2585340, 6249329, 15074649, 36413754, 87877681, 212208719, 512231040, 1236774481, 2985612241, 7208270406, 17401713849, 42012408751
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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G.f.: x*(1+x^2)/((1+x-x^2)*(1-2*x-x^2)). Colin Barker, Nov 13 2012
The following remarks assume the o.g.f. for this sequence is x*(1 + x^2)/((1 + x - x^2)*(1 - 2*x - x^2)).
This sequence is a fourth-order linear divisibility sequence. It is the case P1 = 1, P2 = -2, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy.
exp( Sum_{n >= 1} 3*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Pell(n)*x^n.
exp( Sum_{n >= 1} (-3)*a(n)*x^n/n ) = 1 + Sum_{n >= 1} 3*Fibonacci(n)*(-x)^n. Cf. A002878. (End)
a(n) = Sum_{j=0..n} T(n, j)*A001045(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n.
a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
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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) = 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 A192423(n) = = 2*A192424(n) = (2, 0, 4, 2, 16, ...) and a(n) = (0, 1, 1, 6, 9, ...).
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MATHEMATICA
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LinearRecurrence[{1, 4, -1, -1}, {0, 1, 1, 6}, 40] (* G. C. Greubel, Jul 12 2023 *)
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PROG
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(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x^2)/((1+x-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Jul 12 2023
(SageMath)
@CachedFunction
if (n<4): return (0, 1, 1, 6)[n]
else: return a(n-1) +4*a(n-2) -a(n-3) -a(n-4)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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