A285872 -id:A285872 - cp's OEIS frontend

A285870 a(n) = floor(n/2) - floor((n+1)/6), n >= 0.

0, 0, 1, 1, 2, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 5, 6, 5, 6, 6, 7, 7, 8, 7, 8, 8, 9, 9, 10, 9, 10, 10, 11, 11, 12, 11, 12, 12, 13, 13, 14, 13, 14, 14, 15, 15, 16, 15, 16, 16, 17, 17, 18, 17, 18, 18, 19, 19, 20, 19, 20, 20, 21, 21, 22, 21, 22, 22, 23, 23, 24

Offset: 0

Wolfdieter Lang, May 12 2017

This is the number of integers k in the (left-sided open) interval ((n+1)/6, floor(n/2)]. This sequence is used in A285872(n), the number of zeros of Chebyshev's S(n, x) polynomial (A049310) in the open interval (-sqrt(3), +sqrt(3)).

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A049310, A059169, A285869, A285872.

[Floor(n/2)-Floor((n+1)/6): n in [0..100]]; // Vincenzo Librandi, May 15 2017

Table[Floor[n/2] - Floor[(n + 1)/6], {n, 0, 60}] (* or *)
CoefficientList[Series[(x^2/((1 + x) (1 - x)^2)) (1 - x^3/((1 + x + x^2) (1 - x + x^2))), {x, 0, 60}], x] (* Michael De Vlieger, May 13 2017 *)

concat(vector(2), Vec(x^2*(1 + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^100))) \\ Colin Barker, May 18 2017

a(n) = floor(n/2) - floor((n+1)/6), n >= 0.
G.f.: (x^2/((1+x)*(1-x)^2))*(1-x^3/((1+x+x^2)*(1-x+x^2))).
a(n) = a(n-1) + a(n-6) - a(n-7) for n>6. - Colin Barker, May 18 2017

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.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, May 13 2017

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.

			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.

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).

Cf. A008611(n-1) (1), A285869 (sqrt(2)), A285872 (sqrt(3)).

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

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 *)

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

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)

A283431 a(n) is the number of zeros of the Hermite H(n, x) polynomial in the open interval (-1, +1).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9

Offset: 0

Michel Lagneau, May 16 2017

nonn

The Hermite polynomials satisfy the following recurrence relation:
H(0,x) = 1,
H(1,x) = 2*x,
H(n,x) = 2*x*H(n-1,x) - 2*(n-1)*H(n-2,x).
The first few Hermite polynomials are:
H(0,x) = 1
H(1,x) = 2x
H(2,x) = 4x^2 - 2
H(3,x) = 8x^3 - 12x
H(4,x) = 16x^4 - 48x^2 + 12
H(5,x) = 32x^5 - 160x^3 + 120x

			a(5) = 3 because the zeros of H(5,x) = 32x^5 - 160x^3 + 120x are x1 = -2.0201828..., x2 = -.9585724..., x3 = 0., x4 = .9585724... and x5 = 2.020182... with three roots x2, x3 and x4 in the open interval (-1, +1).

Eric Weisstein's World of Mathematics, Hermite Polynomial.
Index entries for sequences related to Hermite polynomials

Cf. A054373, A054374, A059343, A008611, A096713, A257564, A285872.

for n from 0 to 90 do:it:=0:
y:=[fsolve(expand(HermiteH(n,x)),x,real)]:for m from 1 to nops(y) do:if abs(y[m])<1 then it:=it+1:else fi:od: printf(`%d, `, it):od:

a[n_] := Length@ List@ ToRules@ Reduce[ HermiteH[n, x] == 0 && -1 < x < 1, x]; Array[a, 82, 0] (* Giovanni Resta, May 17 2017 *)

Conjecture: a(n) = A257564(n+2).

cp's OEIS Frontend

A285870 a(n) = floor(n/2) - floor((n+1)/6), n >= 0.

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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.

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Magma

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Formula

A283431 a(n) is the number of zeros of the Hermite H(n, x) polynomial in the open interval (-1, +1).

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