A189089 -id:A189089 - cp's OEIS frontend

A189088 Decimal expansion of Pi - sqrt(Pi^2 - 1).

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

Offset: 0

Clark Kimberling, Apr 16 2011

Decimal expansion of the shape (= length/width = Pi - sqrt(-1+Pi^2)) of the lesser 2*Pi-contraction rectangle.

See A188738 for an introduction to lesser and greater r-contraction rectangles, their shapes, and partitioning these rectangles into a sets of squares in a manner that matches the continued fractions of their shapes.

			0.1634045465204364424868140709760745094117386882...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A188738, A189089, A189090.

r = 2*Pi; t = (r - (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]  (* A189088 *)
ContinuedFraction[t, 120]
RealDigits[Pi-Sqrt[Pi^2-1],10,150][[1]] (* Harvey P. Dale, Sep 25 2016 *)

Pi*(1-sqrt(1-1/Pi^2)) \\ Charles R Greathouse IV, May 07 2011

A189090 Continued fraction of Pi + sqrt(Pi^2 - 1).

Original entry on oeis.org

6, 8, 2, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 14, 2, 1, 6, 3, 1, 1, 2, 3, 7, 1, 1, 3, 2, 2, 1, 3, 10, 1, 1, 4, 3, 3, 10, 1, 2, 2, 18, 3, 77, 1, 1, 18, 1, 2, 2, 4, 1, 2, 8, 1, 4, 1, 44, 1, 28, 1, 4, 1, 1, 2, 116, 1, 1, 2, 2, 1, 5, 4, 5, 27, 4, 1, 3, 1, 3, 5, 1, 2, 2, 1, 16, 1, 3, 1, 5, 2, 1, 1, 25, 3, 1, 1, 17, 1, 5, 3, 1, 2, 1, 4, 12, 4, 7, 42, 19, 1, 2, 23, 1, 3, 2, 1, 4

Offset: 0

Clark Kimberling, Apr 16 2011

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A189089, A189088.

r = 2*Pi; t = (r + (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (*A189089*)
ContinuedFraction[t, 120]  (*A189090*)

contfrac(Pi+sqrt(Pi^2-1)) \\ Charles R Greathouse IV, Jul 29 2011

A359540 Decimal expansion of arccosh(Pi).

Original entry on oeis.org

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

Offset: 1

Wolfe Padawer, Jan 05 2023

			1.811526272460853107021852049305420510220702081057922474861595622974...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 31, page 291.

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A189089.

RealDigits[Abs[ArcCos[Pi]], 10, 105][[1]]

-I*acos(Pi) \\ Hugo Pfoertner, Jan 24 2023

Equals |arccos(Pi)|.

Equals log(Pi + sqrt(Pi^2 - 1)) = log(A189089). - Amiram Eldar, Jan 05 2023

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A189088 Decimal expansion of Pi - sqrt(Pi^2 - 1).

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A189090 Continued fraction of Pi + sqrt(Pi^2 - 1).

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A359540 Decimal expansion of arccosh(Pi).

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