A188725 -id:A188725 - cp's OEIS frontend

A188726 Continued fraction of the shape of a (2*Pi)-extension rectangle; shape = Pi + sqrt(1 + Pi^2).

6, 2, 3, 1, 1, 3, 2, 1, 16, 47, 1, 4, 2, 7, 1, 5, 317, 4, 1, 1, 1, 2, 13, 1, 38, 37, 1, 4, 1, 13, 1, 59, 3, 17, 1, 2, 2, 2, 5, 1, 3, 1, 3, 9, 1, 3, 4, 1, 2, 2, 1, 1, 2, 1, 23, 8, 9, 84, 1, 3, 1, 2, 1, 1, 3, 5, 5, 1, 1, 16, 1, 8, 4, 11, 1, 3, 1, 16, 4, 1, 1, 1, 1, 18, 1, 12, 1, 21, 3, 3, 1, 2, 4, 2, 10, 3, 5, 6, 1, 1, 25, 4, 10, 1, 5, 2, 1, 4, 16, 2, 5, 4, 2, 1, 4, 1, 1, 2, 1, 1

Offset: 0

Clark Kimberling, Apr 09 2011

See A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles of shape r.

A 2*Pi-extension rectangle matches the continued fraction [6,2,3,1,1,3,2,1,16,47,...] of the shape L/W = Pi + sqrt(1 + Pi^2). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (2*Pi)-extension rectangle, 6 squares are removed first, then 2 squares, then 3 squares, then 1 square, then 1 square, ..., so that the original rectangle is partitioned into an infinite collection of squares.

			6.4385009630654083972232325635946917292621665408132...

G. C. Greubel, Table of n, a(n) for n = 0..9999

Cf. A188640, A188725 (decimal expansion).

SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(Pi(R) + Sqrt(1 + Pi(R)^2)); // G. C. Greubel, Oct 31 2018

with(numtheory): cfrac(Pi+sqrt(1+Pi^2),120,'quotients'); # Muniru A Asiru, Nov 22 2018

r = 2*Pi; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A188725 *)
ContinuedFraction[t, 120] (* A188726 *)
ContinuedFraction[Pi + Sqrt[1 + Pi^2], 100] (* G. C. Greubel, Oct 31 2018 *)

default(realprecision, 100); contfrac(Pi + sqrt(1 + Pi^2)) \\ G. C. Greubel, Oct 31 2018

A188720 Decimal expansion of (e+sqrt(4+e^2))/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 09 2011

Decimal expansion of shape of an e-extension rectangle; see A188640 for definitions of shape and r-extension rectangle.  Briefly, an r-extension rectangle is composed of two rectangles having shape r.
An e-extension rectangle matches the continued fraction A188721 of the shape L/W = (1/2) *(e+sqrt(4+e^2)).  This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...].  Specifically, for an e-extension rectangle, 3 squares are removed first, then 21 squares, then 2 squares, then 40 squares, then 1 square,..., so that the original rectangle is partitioned into an infinite collection of squares.
(e+sqrt(4+e^2))/2 = [e,e,e,... ] (continued fraction). - Clark Kimberling, Sep 23 2013

			3.046524695333472471811401766587155243274607058879794774577422496312...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A188640, A188721, A188726, A188722, A188725.

SetDefaultRealField(RealField(100)); (Exp(1) +Sqrt(4+Exp(2)))/2; // G. C. Greubel, Oct 31 2018

evalf((exp(1)+sqrt(4+exp(2)))/2,140); # Muniru A Asiru, Nov 01 2018

r=E; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[(E+Sqrt[4+E^2])/2,10,150][[1]] (* Harvey P. Dale, Jan 07 2015 *)

default(realprecision, 100); (exp(1) + sqrt(4 + exp(2)))/2 \\ G. C. Greubel, Oct 31 2018

A360938 Decimal expansion of arcsinh(Pi).

Original entry on oeis.org

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

Offset: 1

Wolfe Padawer, Feb 26 2023

			1.862295743310848219888361325182620574902674184961554765612879514423...

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

Cf. A188725, A359540, A361518, A361519.

Mathematica
```
RealDigits[ArcSinh[Pi], 10, 105][[1]]
```
PARI
```
asinh(Pi) \\ Michel Marcus, Feb 26 2023
```

Equals log(Pi + sqrt(Pi^2 + 1)) = log(A188725).

Equals Pi*Integral_{x=0..1} 1/sqrt((Pi^2)*x^2 + 1) dx.

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A188726 Continued fraction of the shape of a (2*Pi)-extension rectangle; shape = Pi + sqrt(1 + Pi^2).

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A188720 Decimal expansion of (e+sqrt(4+e^2))/2.

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A360938 Decimal expansion of arcsinh(Pi).

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