A330156 - OEIS

A330156

Decimal expansion of the continued fraction expansion [1; 1/2, 1/3, 1/4, 1/5, 1/6, ...].

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

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OFFSET

1,2

COMMENTS

This constant is formed from the continued fraction [1; 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ...] the reciprocals of the positive integers, A000027.

From Amiram Eldar, Apr 30 2026: (Start)

The aspect ratio of a rectangle for which the expected length of a random chord equals the length of the shorter side, where a random chord is formed by the intersection of the rectangle with a random line whose direction and perpendicular distance from the origin are independently and uniformly distributed. For a rectangle with sides 1 and r, the mean chord length is Pi*r/(2*(r+1)). This constant is the root of the equation Pi*r/(2*(r+1)) = 1.

The harmonic mean of the number of flips of a coin required until the number of heads first exceeds the number of tails (the mean number is infinite; see Young link). (End)

REFERENCES

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.4, p. 23.

LINKS

Table of n, a(n) for n=1..87.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 562.

Wikipedia, Continued Fraction Expansions of Pi.

Alexander Young, A coin is being flipped until more heads than tails have been flipped. What is the expected number of times of flipping the coin?, Quora, 2025.

Index entries for transcendental numbers.

FORMULA

Equals 2 / (Pi - 2).

Equals 1/arctan(cot(1)). - Daniel Hoyt, Apr 11 2020

From Stefano Spezia, Oct 26 2024: (Start)

2/(Pi - 2) = 1 + K_{n>=1} n*(n+1)/1, where K is the Gauss notation for an infinite continued fraction. In the expanded form, 2/(Pi - 2) = 1 + 1*2/(1 + 2*3/(1 + 3*4/(1 + 4*5/(1 + 5*6/(1 + ...))))) (see Finch at p. 23).

2/(Pi - 2) = Sum_{n>=1} (2/Pi)^n (see Shamos). (End)

Equals A309091/2. - Hugo Pfoertner, Oct 28 2024

EXAMPLE

1.7519383938841086612039097015114538792503980068057415636404709501399828870437...

MATHEMATICA

First[RealDigits[2/(Pi - 2), 10, 100]] (* Paolo Xausa, Apr 27 2024 *)

PROG

(PARI) 2 / (Pi - 2) \\ Michel Marcus, Dec 05 2019

(PARI) 1/atan(cotan(1)) \\ Daniel Hoyt, Apr 11 2020

CROSSREFS

Cf. A000027, A000796, A052119, A073824, A309091.

Sequence in context: A154870 A098687 A021137 * A323811 A316334 A196486