A332324 -id:A332324 - cp's OEIS frontend

A117605 Decimal expansion of the real solution to equation x^3 + 3*x = 2.

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

Offset: 0

Zak Seidov, Apr 27 2006

Let p(n,x) denote the n-th Maclaurin polynomial of e^x, and let p'(n,x) denote its derivative. Then p'(n+1,x) = p(n,x), so that the real zero r of p(n,x), for odd n, is also the value of x that minimizes p(n+1,x). Let y = 0.5960716... . Then for n = 3, we have r = - y - 1; see A332324 for the minimal value of p(4,x). - Clark Kimberling, Feb 13 2020

			x = 0.596071637983321523112805414399681828113325...

Cf. A014176, A332324.

RealDigits[N[Solve[3*z+z^3==2,z][[1,1,2]],100]][[1]]

x = (1+sqrt(2))^(1/3) - 1/(1+sqrt(2))^(1/3).
From Gerry Martens, Mar 23 2025: (Start)
Equals (2/3)*hypergeom([1/3, 2/3], [3/2], -1).
Equals 2*sinh(asinh(1)/3). (End)

a(99) corrected by Sean A. Irvine, Jul 25 2021

A383644 a(n) is the number of zeros in the left half-plane of the Maclaurin polynomial of degree n for exp(z).

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 5, 6, 7, 6, 7, 8, 9, 10, 11, 10, 11, 12, 13, 14, 13, 14, 15, 16, 17, 16, 17, 18, 19, 20, 19, 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, 37, 36, 37, 38, 39, 40, 39, 40, 41, 42, 43, 42, 43, 44

Offset: 1

Michel Lagneau, May 03 2025

nonn

The Maclaurin polynomial of degree n for exp(z) is P(n,z) = Sum_{i=0..n} z^i/i!

The number of zeros in the right half-plane is equal to n - a(n) because we do not observe any purely imaginary roots.

			a(4)= 4 because P(4,z) = 1 + z/1! + z^2/2! + z^3/3! + z^4/4! with 4 roots in the left half-plane:
z1 = -1.729444231-.8889743761*i,
z2 = -1.729444231+.8889743761*i,
z3 = -.2705557689-2.504775904*i,
z4 = -.2705557689+2.504775904*i

Cf. A332324, A330187, A332420.

A:=proc(n) local P, m, y, it:
  it:=0:P:=add(x^i/i!,i=0..n):
   y:=[fsolve(expand(P), x, complex)]:
    for m from 1 to nops(y) do:
     if Re(y[m])<0 then
      it:=it+1:else fi:
    od: A(n):=it:end proc:
seq(A(n), n=1..70);

cp's OEIS Frontend

A117605 Decimal expansion of the real solution to equation x^3 + 3*x = 2.

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