A328227 Decimal expansion of positive solution to x^2 = 1 + (Pi + arccos(1/x))^2.

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

Offset: 1

Author

Jack Zhang, Oct 08 2019

Keywords

• nonn
• cons

Comments

We are in a rowboat on a circular lake, starting at the center. At the edge of the lake is a mean goblin. He can run k times as fast as we can row. This is the minimum value of k such that we will not be able to escape.

From Rian Hunter, Jun 16 2021: (Start)

For a spirograph defined by complex function z = p * e^(-i * b * t) + b * e^(i * t), this is the value of p as b->oo such that each petal is tangent to the next one.

If we consider the set of all right triangles such that their tangent value is equal to the opposite angle in radians, this value is equal to the negative secant of the right triangle from that set with the smallest nonzero opposite angle. (End)

The envelope of the t*x = sin(t*y) family of curves contains the set of y = (-1)^n*k_n*x straight lines (n > 0), where k_n is the solution of (n*Pi + arccos(1/k))^2 + 1 = k^2. This entry is k_1. See illustration, section Links. - Luc Rousseau, Mar 11 2022

Maximum negative value of x/sin(x). - Andrew Slattery, Jun 29 2022

Examples

			4.6033388487517003525565820291030165130673971341605323460394301154384587319659...
		

Links

• Rian Hunter, The Number Hiding Inside the Spirograph.
• IBM Research, Ponder This Challenge - May 2001.
• Luc Rousseau, A328227 viewed as the |slope| of an envelope.

Crossrefs

Cf. A115365.

Equals 1/A213053.

Programs

Mathematica

NSolve[x^2==1+(Pi+ArcCos[1/x])^2,x,Reals,WorkingPrecision->100]

PARI

solve(x=4, 5, 1 + (Pi+acos(1/x))^2 - x^2) \\ Michel Marcus, Oct 08 2019

Formula

x=-sec(y), where decimal expansion of y is A115365.

Alternatively, x=sqrt(y^2+1).