cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A332325 Number of Maclaurin polynomials p(2m,x) of cos(x) that have exactly n positive zeros.

Original entry on oeis.org

3, 4, 4, 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, 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
Offset: 1

Views

Author

Clark Kimberling, Feb 11 2020

Keywords

Comments

Maclaurin polynomial p(2m,x) of cos(x) is 1 - x^2/2! + x^4/4! - ... + (-1)^m*x^(2m)/(2m)!.

Examples

			a(1) counts these values of 2m: 2, 6, and 10. The single positive zeros of p(2,x), p(6,x), and p(10,x) are sqrt(2), 1.56990..., and 1.57079..., respectively.

Crossrefs

Cf. A332326, A332420, A358997.

Programs

  • Mathematica
    z = 30; p[m_, x_] := Normal[Series[Cos[x], {x, 0, m }]];
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}]

Extensions

More terms from Jinyuan Wang, Jan 21 2025

A332327 Decimal expansion of the least positive zero of the 6th Maclaurin polynomial of cos x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 11 2020

Keywords

Comments

The Maclaurin polynomial p(2n,x) of cos x is 1 - x^2/2! + x^4/4! + ... + (-1)^n x^(2n)/(2n)!.
Let z(n) be the least positive zero of p(2n,x). The limit of z(n) is Pi/2 = 1.570796326..., as in A019669.

Examples

			Least positive zero = 1.56990582516119145661812218578182974...

Crossrefs

Cf. A019669, A332326.

Programs

  • Mathematica
    z = 150; p[n_, x_] := Normal[Series[Cos[x], {x, 0, n}]]
t = x /. NSolve[p[6, x] == 0, x, z][[4]]
u = RealDigits[t][[1]]
Plot[Evaluate[p[6, x]], {x, -1, 2}]

A332328 Decimal expansion of the least positive zero of the 8th Maclaurin polynomial of cos x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 11 2020

Keywords

Comments

The Maclaurin polynomial p(2n,x) of cos x is 1 - x^2/2! + x^4/4! + ... + (-1)^n x^(2n)/(2n)!.
Let z(n) be the least positive zero of p(2n,x). The limit of z(n) is Pi/2 = 1.570796326..., as in A019669.

Examples

			Least positive zero = 1.5708210679533907291728211531492495531616658...

Crossrefs

Cf. A019669, A332326.

Programs

  • Mathematica
    z = 150; p[n_, x_] := Normal[Series[Cos[x], {x, 0, n}]]
t = x /. NSolve[p[8, x] == 0, x, z][[5]]
u = RealDigits[t][[1]]
Plot[Evaluate[p[8, x]], {x, -1, 2}]
Showing 1-3 of 3 results.

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