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 A286717 a(n) is the number of zeros of the Chebyshev S(n, x) polynomial (A049310) in the open interval (-phi, +phi), with the golden section phi = (1 + sqrt(5))/2. 2
 0, 1, 2, 3, 2, 3, 4, 5, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 14, 15, 16, 17, 18, 17, 18, 19, 20, 21, 20, 21, 22, 23, 24, 23, 24, 25, 26, 27, 26, 27, 28, 29, 30, 29, 30, 31, 32, 33, 32, 33, 34, 35, 36, 35, 36, 37, 38, 39, 38, 39, 40, 41, 42 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS G. C. Greubel, Table of n, a(n) for n = 0..5000 Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1). FORMULA 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) = A286716(n). See the g.f. for {b(n)}_{n>=0} there. From Colin Barker, May 18 2017: (Start) G.f.: x*(1 + x + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)). a(n) = a(n-1) + a(n-5) - a(n-6) for n>5. (End) EXAMPLE a(4) = 2: S(4, x) = 1+x^4-3*x^2, and only two of the four zeros -phi, -1/phi, +1/phi, phi are in the open interval (-phi, +phi), the other two are at the borders. MATHEMATICA CoefficientList[Series[x*(1+x+x^2-x^3+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)), {x, 0, 50}], x] (* G. C. Greubel, Mar 08 2018 *) LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 2, 3}, 80] (* Harvey P. Dale, Aug 20 2020 *) PROG (PARI) concat(0, Vec(x*(1 + x + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^100))) \\ Colin Barker, May 18 2017 (Magma) m:=80; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x+x^2-x^3+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)))); // G. C. Greubel, Mar 08 2018 CROSSREFS Cf. A008611(n-1) (1), A285869 (sqrt(2)), A285872 (sqrt(3)). Sequence in context: A215653 A358503 A327666 * A162751 A026342 A275727 Adjacent sequences: A286714 A286715 A286716 * A286718 A286719 A286720 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, May 13 2017 STATUS approved

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Last modified August 5 11:26 EDT 2024. Contains 374950 sequences. (Running on oeis4.)