A131594 -id:A131594 - cp's OEIS frontend

A102208 Decimal expansion of the volume of an icosahedron with unit edge length.

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 17 2005

			2.181694990624912373503822...

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

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Icosahedron.
Wikipedia, Icosahedron - Area and volume.
Index entries for algebraic numbers, degree 2.

Cf. A001622 (phi), A020829 (regular tetrahedron volume), A131594 (regular octahedron volume), A102769 (regular dodecahedron volume).

Cf. A071402.

RealDigits[5/12*(3+Sqrt[5]),10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

5*(3 + sqrt(5))/12 \\ G. C. Greubel, Jul 06 2017

Equals 5 * (3 + sqrt(5))/12.

Equals 5*phi^2/6, phi being the golden ratio. - Stanislav Sykora, Nov 23 2013

A179587 Decimal expansion of the volume of square cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

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

			1.942809041582063365867792482806465385713114583584632048784453158660...

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

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

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

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 *)

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

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

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

A102769 Decimal expansion of the volume of a dodecahedron with each edge of unit length.

Original entry on oeis.org

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 10 2005

Equals 5*phi^3/(2*xi^2), phi being the golden ratio (A001622) and xi its associate (A182007). - Stanislav Sykora, Nov 23 2013

			7.663118960624631968716053920...

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

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Dodecahedron.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A001622 (phi), A182007 (phi associate), A020829 (regular tetrahedron volume), A131594 (regular octahedron volume), A102208 (regular icosahedron volume).

evalf((15+7*sqrt(5))/4,100); # Wesley Ivan Hurt, Jan 29 2017

RealDigits[(Sqrt[5]/2)*(GoldenRatio)^4, 10, 50][[1]] (* G. C. Greubel, Jul 06 2017 *)

(7*sqrt(5)+15)/4 \\ Charles R Greathouse IV, Apr 25 2016

Equals (15 + 7 sqrt(5)) / 4.

Equals (sqrt(5)/2)*(phi)^4, where phi is the golden ratio. - G. C. Greubel, Jul 06 2017

A020829 Decimal expansion of 1/sqrt(72) = 1/(3*2^(3/2)) = sqrt(2)/12.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Volume of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
In the dragon curve fractal, (5/6)*sqrt(2) = 1.1785.... is the maximum distance of any point from curve start.  Such a maximum must be to a vertex of the convex hull.  Hull vertices are shown by Benedek and Panzone (theorem 3, page 85) and their P8 = 7/6 - (1/6)i at distance sqrt((7/6)^2 + (1/6)^2) is the maximum. - Kevin Ryde, Nov 22 2019
With offset 1, volume of a triangular cupola (Johnson solid J_3) with unit edges. - Paolo Xausa, Aug 04 2025

			0.117851130197757920733474...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Agnes I. Benedek and Rafael Panzone, On Some Notable Plane Sets, II: Dragons, Revista de la Unión Matemática Argentina, volume 39, numbers 1-2, 1994, pages 76-90.
Wikipedia, Platonic solid.
Wikipedia, Tetrahedron.
Wikipedia, Triangular cupola.
Index entries for algebraic numbers, degree 2

Cf. A131594 (regular octahedron volume), A102208 (regular icosahedron volume), A102769 (regular dodecahedron volume).

Cf. A010524, A020765, A020775, A378207.

RealDigits[Sqrt[2]/12, 10, 50][[1]] (* G. C. Greubel, Jul 06 2017 *)

sqrt(2)/12 \\ G. C. Greubel, Jul 06 2017

Equals Integral_{x=0..Pi/4} sin(x)^2 * cos(x) dx. - Amiram Eldar, May 31 2021

Equals 1/A010524 = A020765/3 = A020775/2 = A378207/5. - Hugo Pfoertner, Jan 26 2025

A232811 Decimal expansion of the surface index of a regular octahedron.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Dec 01 2013

Equivalently, the surface area of a regular octahedron with unit volume. Among Platonic solids, surface indices decrease with increasing number of faces: A232812 (tetrahedron), 6.0 (cube = hexahedron), this one, A232810 (dodecahedron), and A232809 (icosahedron).

An algebraic integer of degree 6 with minimal polynomial x^6 - 34992. - Charles R Greathouse IV, Apr 25 2016

			5.7191057579816194425444539723965629466374425679...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Platonic solid
Index entries for algebraic numbers, degree 6

Cf. A010469, A131594, A232808 (surface index for a sphere), A232809, A232810, A232812.

RealDigits[Sqrt[3]Surd[36,3],10,120][[1]] (* Harvey P. Dale, Mar 12 2015 *)

sqrtn(34992,6) \\ Charles R Greathouse IV, Apr 25 2016

sqrt(3)*6^(2/3).

A010469/A131594^(2/3).

A274540 Decimal expansion of exp(sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Jun 27 2016

Define P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(q) = C1 and x(n) = 1 for all other n. We find that C2 = lim_{n -> infinity} P(n) = exp((C1-1)/q).
The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.
Some transform pairs: C1 = A002162 (log(2)) and C2 = A135002 (2/exp(1)); C1 = A016627 (log(4)) and C2 = A135004 (4/exp(1)); C1 = A001113 (exp(1)) and C2 = A234473 (exp(exp(1)-1)).
From Peter Bala, Oct 23 2019: (Start)
The constant is irrational: Henry Cohn gives the following proof in Todd and Vishals Blog - "By the way, here's my favorite application of the tanh continued fraction: exp(sqrt(2)) is irrational.
Consider sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1). If exp(sqrt(2)) were rational, or even in Q(sqrt(2)), then this expression would be in Q(sqrt(2)). However, it is sqrt(2)*tanh(1/sqrt(2)), and the tanh continued fraction shows that this equals [0,1,6,5,14,9,22,13,...]. If it were in Q(sqrt(2)), it would have a periodic simple continued fraction expansion, but it doesn't." (End)

			c = 4.113250378782927517173581815140304502401663943151...

G. C. Greubel, Table of n, a(n) for n = 1..10000
The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).
Todd Trimble and Vishal Lama, Continued fraction for e, Todd and Vishal’s blog 2008/08/04
Index entries for transcendental numbers

Cf. A274541, A274542, A036039, A135002, A135004, A234473.

Cf. A014176, A002193, A010503, A131594, A020765, A010466, A020775.

Digits := 80: evalf(exp(sqrt(2))); # End program 1.
P := proc(n) : if n=0 then 1 else P(n) := expand((1/n)*(add(x(n-k)*P(k), k=0..n-1))) fi; end: x := proc(n): if n=1 then (1 + sqrt(2)) else 1 fi: end: Digits := 49; evalf(P(120)); # End program 2.

First@ RealDigits@ N[Exp[Sqrt@ 2], 80] (* Michael De Vlieger, Jun 27 2016 *)

my(x=exp(sqrt(2))); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016

c = exp(sqrt(2)).

c = lim_{n -> infinity} P(n) with P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k) for n >= 1, and P(0) = 1, with x(1) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.

More terms from Jon E. Schoenfield, Mar 15 2018

A377342 Decimal expansion of the volume of a truncated octahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			11.3137084989847603904135097936775846285573750030...

Eric Weisstein's World of Mathematics, Truncated Octahedron.
Wikipedia, Truncated octahedron.

Cf. A377341 (surface area), A020797 (circumradius/10), A152623 (midradius).
Cf. A131594 (analogous for a regular octahedron).
Cf. A002193, A010466, A010487.

First[RealDigits[8*Sqrt[2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedOctahedron", "Volume"], 10, 100]]

Equals 8*sqrt(2) = 8*A002193 = 4*A010466 = 2*A010487.

A343965 Decimal expansion of 4 + 10*sqrt(2)/3.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 05 2021

Volume of a rhombicuboctahedron with unit edge length.

Apart from the leading digit and offset the same as A131594. - R. J. Mathar, May 07 2021

			8.7140452079103168293389624140323269285655729...

Wikipedia, Rhombicuboctahedron

Cf. A343964 (rhombicuboctahedron surface area).

SetDefaultRealField(RealField(100)); 4+10*Sqrt(2)/3;

RealDigits[N[4 + 10*Sqrt[2]/3, 100]][[1]] (* Wesley Ivan Hurt, Nov 12 2022 *)

A377299 Decimal expansion of the volume of a truncated cube with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			13.599663291074443561074547379645257699991802085...

Eric Weisstein's World of Mathematics, Truncated Cube.
Wikipedia, Truncated cube.

Cf. A377298 (surface area), A294968 (circumradius), A010503 (midradius - 1), A377296 (Dehn invariant, negated).

Cf. A131594.

First[RealDigits[7 + 14*Sqrt[2]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCube", "Volume"], 10, 100]]

Equals 7 + (14/3)*sqrt(2) = 7 + 14*A131594.

A274542 Decimal expansion of exp(sqrt(2)/3).

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Jun 27 2016

Define P(n) = (1/n)*(sum(x(n-k)*P(k), k=0..n-1)), n >= 1 and P(0) =1 with x(3) = (1 + sqrt(2)) and x(n) = 1 for all other n. We find that C2 = limit(P(n), n -> infinity) = exp(sqrt(2)/3).

The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.

			c = 1.6022429972035601509951777722286787585129616829545471874……

G. C. Greubel, Table of n, a(n) for n = 1..10000
The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).
Index entries for transcendental numbers

Cf. A274540, A274541, A014176, A131594, A036039.

SetDefaultRealField(RealField(100)); Exp[Sqrt[2]/3]; // G. C. Greubel, Aug 19 2018

Digits := 85: evalf(exp(sqrt(2)/3)); # End program 1.
P := proc(n) : if n=0 then 1 else P(n) := expand((1/n)*(add(x(n-k)*P(k), k=0..n-1))) fi; end: x := proc(n): if n=3 then (sqrt(2)+1) else 1 fi: end: Digits := 56; evalf(P(120)); # End program 2.

First@ RealDigits@ N[Exp[Sqrt[2]/3], 85] (* Michael De Vlieger, Jun 27 2016 *)

my(x=exp(sqrt(2)/3)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016

c = exp(sqrt(2)/3)

c = limit(P(n), n -> infinity) with P(n) = (1/n)*(sum(x(n-k)*P(k), k=0..n-1)) for n >= 1, and P(0) =1, with x(3) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.

cp's OEIS Frontend

A102208 Decimal expansion of the volume of an icosahedron with unit edge length.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A179587 Decimal expansion of the volume of square cupola with edge length 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A102769 Decimal expansion of the volume of a dodecahedron with each edge of unit length.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A020829 Decimal expansion of 1/sqrt(72) = 1/(3*2^(3/2)) = sqrt(2)/12.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A232811 Decimal expansion of the surface index of a regular octahedron.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A274540 Decimal expansion of exp(sqrt(2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A377342 Decimal expansion of the volume of a truncated octahedron with unit edge length.

Views