A010527 -id:A010527

A179452

Decimal expansion of sqrt(5 + 2*sqrt(5))/2, the height of a regular pentagon and midradius of an icosidodecahedron with side length 1.

+10
22

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

Height of a regular pentagon with side length 1. - Jared Kish, Oct 16 2014

Volume of a regular decagonal prism with unit side length and height 2. - Wesley Ivan Hurt, May 04 2021

LINKS

Table of n, a(n) for n=1..105.

Eric Weisstein's World of Mathematics, Icosidodecahedron

Eric Weisstein's World of Mathematics, Pentagon

Index entries for algebraic numbers, degree 4

FORMULA

Equals sqrt(5+2*sqrt(5))/2.

EXAMPLE

1.53884176858762670128514528801845491200335107176889621351957812518743...

MAPLE

sqrt(5+2*sqrt(5.))/2

MATHEMATICA

RealDigits[Sqrt[5+2Sqrt[5]]/2, 10, 120][[1]] (* Harvey P. Dale, Jun 23 2017 *)

PROG

(PARI) sqrt(5+2*sqrt(5))/2

CROSSREFS

Cf. A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451.

KEYWORD

nonn,cons,easy,changed

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

EXTENSIONS

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

Edited by M. F. Hasler, Oct 16 2014

STATUS

approved

A179587

Decimal expansion of the volume of square cupola with edge length 1.

+10
20

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Square cupola: 12 vertices, 20 edges, and 10 faces.

Also, decimal expansion of 1 + Product_{n>0} (1-1/(4*n+2)^2). - Bruno Berselli, Apr 02 2013

Decimal expansion of 1 + (least possible ratio of the side length of one inscribed square to the side length of another inscribed square in the same non-obtuse triangle). - L. Edson Jeffery, Nov 12 2014

2*sqrt(2)/3 is the radius of the base of the maximum-volume right cone inscribed in a unit-radius sphere. - Amiram Eldar, Sep 25 2022

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

Victor Oxman and Moshe Stupel, Why are the side lengths of the squares inscribed in a triangle so close to each other?, Forum Geometricorum, Vol. 13 (2013), 113-115.

Wolfram Alpha, Johnson solid 4

Index entries for algebraic numbers, degree 2

FORMULA

Equals (3 + 2*sqrt(2))/3.

Equals 1 + 2*A131594. - L. Edson Jeffery, Nov 12 2014

EXAMPLE

1.942809041582063365867792482806465385713114583584632048784453158660...

MATHEMATICA

RealDigits[N[1+(2*Sqrt[2])/3, 200]]

(* From the second comment: *) RealDigits[N[1 + Product[1 - 1/(4 n + 2)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

PROG

(PARI) sqrt(8)/3+1 \\ Charles R Greathouse IV, Nov 14 2016

CROSSREFS

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A224268.

Cf. A131594 (decimal expansion of sqrt(2)/3).

KEYWORD

nonn,cons,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

STATUS

approved

A179449

Decimal expansion of the volume of great icosahedron with edge length 1.

+10
19

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Great icosahedron: 20 faces, 12 vertices, and 30 edges.

LINKS

Table of n, a(n) for n=1..105.

Eric Weisstein's World of Mathematics, Great Icosahedron

Wikipedia, Great Icosahedron

Index entries for algebraic numbers, degree 2

FORMULA

Digits of 5/12 * (3-sqrt(5)).

EXAMPLE

0.31830500937508762649617763802863490189974235016186428155379281441228294...

MATHEMATICA

RealDigits[N[5*(Sqrt[5]-3)/12, 105]][[1]]

CROSSREFS

Cf. A010527, A102208, A179290, A179292, A179294.

KEYWORD

nonn,cons,easy,changed

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

EXTENSIONS

Partially rewritten by Charles R Greathouse IV, Feb 02 2011

STATUS

approved

A179450

Decimal expansion of the volume of an icosidodecahedron with edge length 1.

+10
17

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

2,2

COMMENTS

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

LINKS

Table of n, a(n) for n=2..106.

Index entries for algebraic numbers, degree 2

FORMULA

(45 + 17*sqrt(5))/6.

EXAMPLE

13.83552593624940413982599206140528266708175201889932288543420886199647...

MATHEMATICA

RealDigits[N[(45+17*Sqrt[5])/6, 200]]

PROG

(PARI) (45 + 17*sqrt(5))/6 \\ Charles R Greathouse IV, Oct 30 2023

CROSSREFS

Cf. A010527, A102208, A179290, A179292, A179294, A179449.

KEYWORD

nonn,cons,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

EXTENSIONS

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

STATUS

approved

A179451

Decimal expansion of the surface area of an icosidodecahedron with side length 1.

+10
17

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

LINKS

Table of n, a(n) for n=2..106.

Index entries for algebraic numbers, degree 8

FORMULA

Sqrt(30*(10+3*sqrt(5)+sqrt(75+30*sqrt(5))))

EXAMPLE

29.3059828449119895407453754361926770276025163091742830907641713815460...

MATHEMATICA

RealDigits[N[Sqrt[30*(10+3*Sqrt[5]+Sqrt[75+30*Sqrt[5]])], 200]]

PROG

(PARI) polrootsreal(x^8 - 1200*x^6 + 324000*x^4 - 27000000*x^2 + 324000000)[8] \\ Charles R Greathouse IV, Oct 30 2023

CROSSREFS

Cf. A010527, A102208, A179290, A179292, A179294, A179449, A179450.

KEYWORD

nonn,cons,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

EXTENSIONS

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

STATUS

approved

A179552

Decimal expansion of the volume of pentagonal pyramid with edge length 1.

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Pentagonal pyramid: 6 faces, 6 vertices, and 10 edges.

LINKS

Table of n, a(n) for n=0..104.

Wolfram Alpha, Johnson solid 2

Index entries for algebraic numbers, degree 2

FORMULA

Digits of (5+sqrt(5))/24.

EXAMPLE

0.3015028323958245706837155695304698431433590983171469051779540518921...

MATHEMATICA

RealDigits[N[(5+Sqrt[5])/24, 200]]

PROG

(PARI) (5+sqrt(5))/24 \\ Charles R Greathouse IV, Oct 30 2023

CROSSREFS

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452.

KEYWORD

nonn,cons,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

STATUS

approved

A010502

Decimal expansion of square root of 48.

+10
14

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

sqrt(48)/10 is the area enclosed by Koch's fractal snowflake based on unit-sided equilateral triangle (actually 8/5 times the latter's area). - Lekraj Beedassy, Jan 06 2005

7+sqrt(48) is the ratio of outer to inner Soddy circles' radii for three identical kissing circles (see Soddy circles link). - Lekraj Beedassy, Feb 14 2006

Continued fraction expansion is 6 followed by {1, 12} repeated. - Harry J. Smith, Jun 06 2009

Let a, b, c the sides of a triangle ABC of area S, then 4*sqrt(3) <= (a^2+b^2+c^2) / S; equality is obtained only when the triangle is equilateral (see Mitrinovic reference). - Bernard Schott, Sep 27 2022

REFERENCES

J. N. Kapur, Mathematics Enjoyment For The Millions, Problem 47 pp. 64-67, Arya Book Depot, New Delhi 2000.

D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.3, page 112.

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..20000

L. Riddle, Area of the Koch Snowflake

Bernard Schott, Soddy circles

Index entries for algebraic numbers, degree 2

FORMULA

Equals 4*A002194. - R. J. Mathar, Jul 31 2010

Equals A176053/A246724 - 7 (2nd comment and Soddy link). - Bernard Schott, Mar 17 2022

Equals 1/A020805. - Bernard Schott, Sep 28 2022

EXAMPLE

6.928203230275509174109785366023489467771221015241522512223227917807732...

MATHEMATICA

RealDigits[N[Sqrt[48], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2011 *)

PROG

(PARI) default(realprecision, 20080); x=sqrt(48); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010502.txt", n, " ", d)); \\ Harry J. Smith, Jun 06 2009

CROSSREFS

Cf. A040041 (continued fraction).

Cf. A002194, A104956, A010527, A152623 (other geometric inequalities).

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

STATUS

approved

A131595

Decimal expansion of 3*(sqrt(25 + 10*sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.

+10
14

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

Surface area of a regular dodecahedron: A = 3*(sqrt(25 + 10*sqrt(5)))* a^2, where 'a' is the edge.

REFERENCES

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

LINKS

G. C. Greubel, Table of n, a(n) for n = 2..10001

Eric Weisstein's World of Mathematics, Dodecahedron.

Wikipedia, Platonic solid.

Index entries for algebraic numbers, degree 4.

FORMULA

From Stanislav Sykora, Nov 30 2013: (Start)

Equals 15/tan(Pi/5).

Equals 15*phi/xi, where phi is the golden ratio (A001622) and xi its associate (A182007). (End)

EXAMPLE

20.64572880706760307310814372866331519288849004012237995...

MAPLE

evalf(3*(sqrt(25+10*sqrt(5))), 130); # Muniru A Asiru, Nov 02 2018

MATHEMATICA

RealDigits[3*Sqrt[25+10*Sqrt[5]], 10, 120][[1]] (* Harvey P. Dale, Jun 21 2011 *)

PROG

(PARI) default(realprecision, 100); 3*(sqrt(25 + 10*sqrt(5))) \\ G. C. Greubel, Nov 02 2018

(Magma) SetDefaultRealField(RealField(100)); 3*(Sqrt(25 + 10*Sqrt(5))); // G. C. Greubel, Nov 02 2018

CROSSREFS

Cf. A102769, A001622 (phi), A182007 (associate of phi), A010527 (icosahedron/10), A010469 (octahedron), A002194 (tetrahedron). - Stanislav Sykora, Nov 30 2013

KEYWORD

nonn,cons,easy,changed

AUTHOR

Omar E. Pol, Aug 30 2007

EXTENSIONS

More terms from Harvey P. Dale, Jun 21 2011

STATUS

approved

A144981

Decimal expansion of cos(Pi/8) = cos(22.5 degrees).

+10
12

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Also the real part of i^(1/4). - Stanislav Sykora, Apr 25 2012

Width of a regular octagon of unit diameter. See Bingane and Audet. - Michel Marcus, Oct 04 2021

Minimal polynomal 8x^4 - 8x^2 + 1. - Charles R Greathouse IV, Oct 30 2023

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

Christian Bingane and Charles Audet, The equilateral small octagon of maximal width, arXiv:2110.00036 [math.MG], 2021.

Wikipedia, Exact trigonometric constants.

Index entries for algebraic numbers, degree 4

FORMULA

Equals sqrt(2 + sqrt(2))/2 = sqrt(3.41421...)/2 = 1.8477759.../2.

Equals Hypergeometric2F1([11/16, 5/16], [1/2], 3/4) / 2. - R. J. Mathar, Oct 27 2008

EXAMPLE

Equals 0.923879532511286756128183189396788286822416625863642486115097...

MAPLE

evalf(sqrt(2+sqrt(2))/2) ;

MATHEMATICA

RealDigits[ Sqrt[2 + Sqrt[2]]/2, 10, 111][[1]] (* Or *) RealDigits[ Cos[Pi/8], 10, 111][[1]] (* Robert G. Wilson v *)

PROG

(PARI) cos(Pi/8) \\ Michel Marcus, Dec 15 2015

(SageMath) numerical_approx(sqrt(2+sqrt(2))/2, digits=120) # G. C. Greubel, Sep 04 2022

CROSSREFS

Cf. A019863: cos(Pi/5); A010527: cos(Pi/6); A073052: cos(Pi/7); A019879: cos(Pi/9).

KEYWORD

cons,easy,nonn

AUTHOR

R. J. Mathar, Sep 28 2008

STATUS

approved

A179553

Decimal expansion of the surface area of pentagonal pyramid with edge length 1.

+10
11

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Pentagonal pyramid: 6 faces, 6 vertices, and 10 edges.

LINKS

Table of n, a(n) for n=1..105.

Wolfram Alpha, Johnson solid 2

Index entries for algebraic numbers, degree 8

FORMULA

Digits of sqrt(5/2*(10+sqrt(5)+sqrt(75+30sqrt(5))))/2.

EXAMPLE