A332331 Decimal expansion of the next-to-least positive zero of the 12th Maclaurin polynomial of cos x.

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

Offset

1,1

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 next-to-least positive zero of p(2n,x) if there is such a zero. The limit of z(n) is 3 Pi/2 = 4.7123889..., as in A197723.

Links

Table of n, a(n) for n=1..86.

Index entries for algebraic numbers, degree 12.

Example

Next-to-least positive zero = 4.6865176637957574465700489837907750...

Mathematica

z = 150; p[n_, x_] := Normal[Series[Cos[x], {x, 0, n}]]
t = x /. NSolve[p[12, x] == 0, x, z][[8]]
u = RealDigits[t][[1]]
Plot[Evaluate[p[12, x]], {x, -1, 5}]

Prog

(PARI) polrootsreal(x^12 - 132*x^10 + 11880*x^8 - 665280*x^6 + 19958400*x^4 - 239500800*x^2 + 479001600)[4] \\ Charles R Greathouse IV, Jul 06 2026

Crossrefs

Cf. A197723, A332329, A332330.

Sequence in context: A244850 A188930 A396517 * A352421 A159576 A199626

Adjacent sequences: A332328 A332329 A332330 * A332332 A332333 A332334

Keyword

nonn,cons,easy,changed

Author

Clark Kimberling, Feb 11 2020

Status

approved