A173201 Iteration x := x - (sin(x) - cos(x)*x - Pi/2)/(sin(x)*x).

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

Offset

1,2

Comments

Decimal expansion of psi, the unique solution on (0,Pi) of sin(psi) - psi*cos(psi) = Pi/2, an auxiliary constant used in the Hall-Tenenbaum inequality applied to real multiplicative functions. - Jean-François Alcover, Sep 05 2014

Links

Table of n, a(n) for n=1..105.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 28.

M. Fraser, A tale of two goats, Math. Mag., 55 (1982), 221-227.

Gerd Lamprecht, Iterationsrechner mit Algorithmus

Gerd Lamprecht, 10000 digits

Gerd Lamprecht, Zahlenfolgen (sequence)

Formula

x := x - (sin(x) - cos(x)*x - Pi/2)/(sin(x)*x).

Equals 2*arccos(A133731/2).

Example

A133731 = cos(1.9056957293.../2)*2.

Mathematica

psi = x /. FindRoot[Sin[x] - x*Cos[x] == Pi/2, {x, 2}, WorkingPrecision -> 102]; RealDigits[psi] // First (* Jean-François Alcover, Sep 05 2014 *)

Prog

(Other) Gerd Lamprecht Iterationsrechner: #(sin(x)-cos(x)*x-PI/2)/(sin(x)*x)@Na=2; @Nb=a; a=a-Fx(a); @N@Aa-b)%3C%204e-16@N0@N1@Nc=cos(a/2)*2;
(PARI) solve(x=1, 2, sin(x)-x*cos(x)-Pi/2) \\ Charles R Greathouse IV, Mar 03 2021

Crossrefs

Cf. A072112, A133731.

Sequence in context: A195483 A093070 A010533 * A345158 A019740 A372270

Adjacent sequences: A173198 A173199 A173200 * A173202 A173203 A173204

Keyword

cons,nonn

Author

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 12 2010

Status

approved