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A285872
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a(n) is the number of zeros of the Chebyshev S(n, x) polynomial (A049310) in the open interval (-sqrt(3), +sqrt(3)).
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4
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0, 1, 2, 3, 4, 3, 4, 5, 6, 7, 8, 7, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 15, 16, 17, 18, 19, 20, 19, 20, 21, 22, 23, 24, 23, 24, 25, 26, 27, 28, 27, 28, 29, 30, 31, 32, 31, 32, 33, 34, 35, 36, 35, 36, 37, 38, 39, 40, 39, 40, 41, 42, 43, 44, 43, 44, 45
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OFFSET
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0,3
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COMMENTS
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See a May 06 2017 comment on A049310 where these problems are considered which originated in a conjecture by Michel Lagneau (see A008611) on Fibonacci polynomials.
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LINKS
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FORMULA
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a(n) = 2*b(n) if n is even and 1 + 2*b(n) if n is odd with b(n) = floor(n/2) - floor((n+1)/6) = A285870(n). See the g.f. for {b(n)}_{n>=0} there.
G.f.: x*(1 + x + x^2 + x^3 - x^4 + x^5) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-6) - a(n-7) for n>6.
(End)
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EXAMPLE
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n = 3: S(3, x) = x*(-2 + x^2), with all three zeros (-sqrt(2), 0, +sqrt(2)) in the interval (-sqrt(3), +sqrt(3)).
n = 4: S(4, x) = 1 - 3*x^2 + x^4, all four zeros (-phi, -1/phi, 1/phi, phi) with phi = (1 + sqrt(5))/2, approximately 1.618, lie in the interval.
n = 6, two zeros of S(6, x) = -1 + 6*x^2 - 5*x^4 + x^6 are out of the interval (-sqrt(3), +sqrt(3)), namely - 1.8019... and +1.8019... .
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MATHEMATICA
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CoefficientList[Series[x*(1+x+x^2+x^3-x^4+x^5)/((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)), {x, 0, 50}], x] (* G. C. Greubel, Mar 08 2018 *)
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PROG
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(PARI) concat(0, Vec(x*(1 + x + x^2 + x^3 - x^4 + x^5) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^100))) \\ Colin Barker, May 18 2017
(Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x+x^2+x^3-x^4+x^5)/((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)))); // G. C. Greubel, Mar 08 2018
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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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