

A328227


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


1



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
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OFFSET

1,1


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


LINKS

Table of n, a(n) for n=1..95.
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.


FORMULA

x=sec(y), where decimal expansion of y is A115365.
Alternatively, x=sqrt(y^2+1).


EXAMPLE

4.6033388487517003525565820291030165130673971341605323460394301154384587319659...


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A115365.
Equals 1/A213053.
Sequence in context: A021960 A096256 A319091 * A059750 A243983 A117036
Adjacent sequences: A328224 A328225 A328226 * A328228 A328229 A328230


KEYWORD

nonn,cons


AUTHOR

Jack Zhang, Oct 08 2019


STATUS

approved



