A020775 -id:A020775 - cp's OEIS frontend

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

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

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

A377344 Decimal expansion of the volume of a truncated cuboctahedron (great rhombicuboctahedron) with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 26 2024

			41.798989873223330683223642138935773099975406255...

Eric Weisstein's World of Mathematics, Great Rhombicuboctahedron.
Wikipedia, Truncated cuboctahedron.

Cf. A377343 (surface area), A377345 (circumradius), A377346 (midradius).
Cf. A020775 (analogous for a cuboctahedron, with offset 1).
Cf. A002193.

First[RealDigits[22 + 14*Sqrt[2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCuboctahedron", "Volume"], 10, 100]]

Equals 22 + 14*sqrt(2) = 22 + 14*A002193.

A343199 Decimal expansion of 6+2*sqrt(3).

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 07 2021

Decimal expansion of the surface area of a cuboctahedron with unit edge length.

Essentially the same sequence of digits as A176394 and A010469. - R. J. Mathar, Jun 10 2021

			9.4641016151377545870548926830117447338856...

Wikipedia, Cuboctahedron

Cf. A020775 (cuboctahedron volume).

SetDefaultRealField(RealField(200)); 6+2*Sqrt(3);

RealDigits[N[6 + 2*Sqrt[3], 100]][[1]] (* Wesley Ivan Hurt, Nov 12 2022 *)

A386739 Decimal expansion of the volume of a sphenocorona with unit edges.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 01 2025

The sphenocorona is Johnson solid J_86.

			1.5153516399764065597284793124718129048228695068...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Sphenocorona.

Cf. A010482 (surface area - 2), A178809 (surface area + 4).

Cf. A010507, A020775, A115754, A386740, A386741.

First[RealDigits[Sqrt[1 + 3*Sqrt[3/2] + Sqrt[13 + Sqrt[54]]]/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J86", "Volume"], 10, 100]]

Equals sqrt(1 + 3*sqrt(3/2) + sqrt(13 + 3*sqrt(6)))/2 = sqrt(1 + 3*A115754  + sqrt(13 + A010507))/2.
Equals A386740 - A020775.
Equals the largest real root of 1024*x^8 - 1024*x^6 - 3008*x^4 - 96*x^2 + 9.

A386740 Decimal expansion of the volume of an augmented sphenocorona with unit edges.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 01 2025

The augmented sphenocorona is Johnson solid J_87.

			1.7510539003719224011954274331734292512511481527...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Augmented sphenocorona.

Cf. A010502 (surface area - 1).

Cf. A010507, A020775, A115754, A386739, A386741.

First[RealDigits[Sqrt[1 + 3*Sqrt[3/2] + Sqrt[13 + Sqrt[54]]]/2 + 1/Sqrt[18], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J87", "Volume"], 10, 100]]

Equals sqrt(1 + 3*sqrt(3/2) + sqrt(13 + 3*sqrt(6)))/2 + 1/(3*sqrt(2)) = sqrt(1 + 3*A115754  + sqrt(13 + A010507))/2 + A020775.
Equals A386739 + A020775.
Equals the largest real root of 45137758519296*x^16 - 110336743047168*x^14 - 191069246324736*x^12 + 209269081571328*x^10 + 364547659290624*x^8 - 58793017190400*x^6 + 3306865979520*x^4 - 1275399855936*x^2 + 1439671249.

A381685 Decimal expansion of the isoperimetric quotient of a cuboctahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 04 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.74121070749643846336987262839110414827328532846...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A020775 (volume, with offset 1), A343199 (surface area).

Cf. A000796, A002194, A381684.

First[RealDigits[25*Pi/(3 + Sqrt[3])^3, 10, 100]]

Equals 3600*Pi*A020775^2/(A343199^3).

Equals 25*Pi/((3 + sqrt(3))^3) = 25*A000796/((3 + A002194)^3).

A386435 Decimal expansion of the largest dihedral angle, in radians, in a triangular bipyramid (Johnson solid J_12).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 20 2025

Also the largest dihedral angle in a triangular orthobicupola (Johnson solid J_27) and the second largest dihedral angle in an augmented truncated tetrahedron (Johnson solid J_65).

			2.461918834681549364269858356495974751420680018710...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Augmented truncated tetrahedron.
Wikipedia, Triangular bipyramid.
Wikipedia, Triangular orthobicupola.

Cf. A137914 (J_12 smallest dihedral angle).

Cf. A020775 (J_12 volume), A104956 (J_12 surface area).

First[RealDigits[ArcCos[-7/9], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J12", "DihedralAngles"]], 10, 100]]

Equals arccos(-7/9).

cp's OEIS Frontend

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

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

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A377344 Decimal expansion of the volume of a truncated cuboctahedron (great rhombicuboctahedron) with unit edge length.

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A343199 Decimal expansion of 6+2*sqrt(3).

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A386739 Decimal expansion of the volume of a sphenocorona with unit edges.

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A386740 Decimal expansion of the volume of an augmented sphenocorona with unit edges.

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A381685 Decimal expansion of the isoperimetric quotient of a cuboctahedron.

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