A012265 -id:A012265 - cp's OEIS frontend

A012264 Number of real roots of x - x^3/3! + x^5/5! - ... + (-1)^n*x^(2n+1)/(2n+1)!.

1, 3, 1, 3, 5, 3, 5, 7, 5, 7, 5, 7, 9, 7, 9, 7, 9, 11, 9, 11, 13, 11, 13, 11, 13, 15, 13, 15, 17, 15, 17, 15, 17, 19, 17, 19, 17, 19, 21, 19, 21, 23, 21, 23, 21, 23, 25, 23, 25, 23, 25, 27, 25, 27, 29, 27, 29, 27, 29, 31, 29, 31, 33

Offset: 0

N. J. A. Sloane

James Propp, posting to math-fun mailing list May 30 1997.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A012265.

Digits := 25: t1 := 0: for k from 1 by 2 to 51 do t1 := t1+(-1)^( (k-1)/2 )*x^k/k!; print(nops([ fsolve(t1*k!) ])); od:

f[n_] := Sum[ (-1)^k*x^(2k+1)/(2k+1)!, {k, 0, n}]; a[n_] := CountRoots[f[n], x]; Table[a[n], {n, 0, 62}] (* Jean-François Alcover, Sep 13 2012 *)

More terms from James Sellers

A332420 Number of Maclaurin polynomials p(2m-1,x) of sin(x) having exactly n positive zeros.

Original entry on oeis.org

3, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4

Offset: 1

Clark Kimberling, Feb 13 2020

nonn

Maclaurin polynomial p(2m-1,x) of sin(x) is x - x^3/3! + x^5/5! - ... - (-1)^m*x^(2m-1)/(2m-1)!.

			a(1) counts these values of 2m-1: 3, 5, and 11. The single zeros of p(3,x), p(5,x), and p(11,x) are sqrt(6), 3.078642..., and 3.141148..., respectively.

Cf. A012265, A332325.

z = 60; p[n_, x_] := Normal[Series[Sin[x], {x, 0, n}]];
t[n_] := x /. NSolve[p[n, x] == 0, x, z];
u[n_] := Select[t[n], Im[#] == 0 && # > 0 &];
v = Table[Length[u[n]], {n, 2, 100, 2}]
Table[Count[v, n], {n, 1, 10}]

More terms from Jinyuan Wang, Jan 21 2025

A358997 a(n) is the number of distinct positive real roots of the Maclaurin polynomial of degree 2*n for cos(x).

Original entry on oeis.org

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

Offset: 0

Robert Israel, Dec 09 2022

It appears that a(n) == n (mod 2) and a(n+2) - a(n) is always either 0 or 2.

			a(2) = 2 because the Maclaurin polynomial of degree 4, 1 - x^2/2! + x^4/4!, has two distinct nonnegative real roots, namely sqrt(6-2*sqrt(3)) and sqrt(6+2*sqrt(3)).

Robert Israel, Table of n, a(n) for n = 0..250

Cf. A012265, A332325.

f:= proc(n) local p, k;
  p:= add((-1)^k * x^k/(2*k)!, k=0..n);
  sturm(sturmseq(p,x),x,0,infinity)
end proc:
map(f, [$0..100]);

a[n_] := CountRoots[Sum[(-1)^k*x^k/(2k)!, {k, 0, n}], {x, 0, Infinity}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 12 2023 *)

cp's OEIS Frontend

A012264 Number of real roots of x - x^3/3! + x^5/5! - ... + (-1)^n*x^(2n+1)/(2n+1)!.

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A332420 Number of Maclaurin polynomials p(2m-1,x) of sin(x) having exactly n positive zeros.

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A358997 a(n) is the number of distinct positive real roots of the Maclaurin polynomial of degree 2*n for cos(x).

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Keywords

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Examples

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Programs

Maple

Mathematica