A381684 -id:A381684 - cp's OEIS frontend

A382009 Decimal expansion of the isoperimetric quotient of a triakis icosahedron.

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

Offset: 0

Paolo Xausa, Mar 20 2025

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

			0.9051808017402297976528502417935535833001760067...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A378973 (surface area), A378974 (volume).

Cf. A000796, A002163, A381684.

First[RealDigits[Pi/183*Sqrt[(169753 + 75771*Sqrt[5])/122], 10, 100]]

Equals 36*Pi*A378974^2/(A378973^3).

Equals (Pi/183)*sqrt((169753 + 75771*sqrt(5))/122) = (A000796/183)*sqrt((169753 + 75771*A002163)/122).

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).

A381686 Decimal expansion of the isoperimetric quotient of a truncated cube.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 04 2025

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

			0.61302821107928032110240558144714079708976169223933...

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

Cf. A377298 (surface area), A377299 (volume).

Cf. A000796, A002194, A010524, A156164, A381684.

First[RealDigits[49*Pi*(17 + 12*Sqrt[2])/(2*(6 + 6*Sqrt[2] + Sqrt[3])^3), 10, 100]]

Equals 36*Pi*A377299^2/(A377298^3).

Equals 49*Pi*(17 + 12*sqrt(2))/(2*(6 + 6*sqrt(2) + sqrt(3))^3) = 49*A000796*A156164/(2*(6 + A010524 + A002194)^3).

A381687 Decimal expansion of the isoperimetric quotient of a truncated octahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 04 2025

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

			0.753366625166156882229489414578751361927704595866...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377341 (surface area), A377342 (volume).

Cf. A000796, A010469, A381684.

First[RealDigits[64*Pi/(3*(1 + 2*Sqrt[3])^3), 10, 100]]

Equals 36*Pi*A377342^2/(A377341^3).

Equals 64*Pi/(3*(1 + 2*sqrt(3))^3) = 64*A000796/(3*(1 + A010469)^3).

A381688 Decimal expansion of the isoperimetric quotient of a (small) rhombicuboctahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 04 2025

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

			0.8684680457998683969444644014889968511341210788731...

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

Cf. A343964 (surface area), A343965 (volume).

Cf. A000796, A002193, A002194, A381684.

First[RealDigits[4*Pi*(43 + 30*Sqrt[2])/(9 + Sqrt[3])^3, 10, 100]]

Equals 36*Pi*A343965^2/(A343964^3).

Equals 4*Pi*(43 + 30*sqrt(2))/((9 + sqrt(3))^3) = 4*A000796*(43 + 30*A002193)/((9 + A002194)^3).

A381689 Decimal expansion of the isoperimetric quotient of a truncated cuboctahedron (great rhombicuboctahedron).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 06 2025

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

			0.8390038051045342803688923433479361567507803475...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377343 (surface area), A377344 (volume).

Cf. A002193, A002194, A010549, A019679, A381684.

First[RealDigits[Pi/12*(11 + Sqrt[98])^2/(2 + Sqrt[2] + Sqrt[3])^3, 10, 100]]

Equals 36*Pi*A377344^2/(A377343^3).

Equals (Pi/12)*(11 + 7*sqrt(2))^2/((2 + sqrt(2) + sqrt(3))^3) = A019679*(11 + A010549)^2/((2 + A002193 + A002194)^3).

A381690 Decimal expansion of the isoperimetric quotient of a snub cube (snub cuboctahedron).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 06 2025

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

			0.8991805073406237668693240204602002492109710905019...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377602 (surface area), A377603 (volume).

Cf. A000796, A010502, A381684.

First[RealDigits[Pi/(3 + Sqrt[48])^3*Root[4*#^3 - 1128*#^2 + 2154*# - 6241 &, 1], 10, 100]]

Equals 36*Pi*A377603^2/(A377602^3).

Equals (Pi/((3 + 4*sqrt(3))^3))*r = (A000796/((3 + A010502)^3))*r, where r is the real root of 4*x^3 - 1128*x^2 + 2154*x - 6241.

A381691 Decimal expansion of the isoperimetric quotient of an icosidodecahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 07 2025

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

			0.86015129192900573693498871755110480766816425346...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A179451 (surface area), A179450 (volume).

Cf. A000796, A002163, A010464, A381684.

First[RealDigits[Pi*(765 + 347*#)/(15*Sqrt[6]*(10 + 3*# + Sqrt[75 + 30*#])^(3/2)) & [Sqrt[5]], 10, 100]]

Equals 36*Pi*A179450^2/(A179451^3).

Equals Pi*(765 + 347*sqrt(5))/(15*sqrt(6)*(10 + 3*sqrt(5) + sqrt(75 + 30*sqrt(5)))^(3/2)) = A000796*(765 + 347*A002163)/(15*A010464*(10 + 3*A002163 + sqrt(75 + 30*A002163))^(3/2)).

A381692 Decimal expansion of the isoperimetric quotient of a truncated dodecahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 07 2025

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

			0.7940548943037979813018428272822581808271192993785...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377694 (surface area), A377695 (volume).

Cf. A000796, A002163, A002194, A010476, A381684.

First[RealDigits[Pi/20*(99 + 47*Sqrt[5])^2/(Sqrt[3] + 6*Sqrt[5 + Sqrt[20]])^3, 10, 100]]

Equals 36*Pi*A377695^2/(A377694^3).

Equals (Pi/20)*(99 + 47*sqrt(5))^2/((sqrt(3) + 6*sqrt(5 + 2*sqrt(5)))^3) = (A000796/20)*(99 + 47*A002163)^2/((A002194 + 6*sqrt(5 + A010476))^3).

A381693 Decimal expansion of the isoperimetric quotient of a truncated icosahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 08 2025

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

			0.90317079253486933111536917369134449479629558352448...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377750 (surface area), A377751 (volume).

Cf. A019679, A002163, A002194, A010476, A381684.

First[RealDigits[Pi/12*(125 + 43*Sqrt[5])^2/(Sqrt[300] + Sqrt[25 + Sqrt[500]])^3, 10, 100]]

Equals 36*Pi*A377751^2/(A377750^3).

Equals (Pi/12)*(125 + 43*sqrt(5))^2/((10*sqrt(3) + sqrt(5*(5 + 2*sqrt(5))))^3) = A019679*(125 + 43*A002163)^2/((10*A002194 + sqrt(5*(5 + A010476)))^3).

cp's OEIS Frontend

A382009 Decimal expansion of the isoperimetric quotient of a triakis icosahedron.

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A381685 Decimal expansion of the isoperimetric quotient of a cuboctahedron.

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A381686 Decimal expansion of the isoperimetric quotient of a truncated cube.

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A381687 Decimal expansion of the isoperimetric quotient of a truncated octahedron.

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A381688 Decimal expansion of the isoperimetric quotient of a (small) rhombicuboctahedron.

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A381689 Decimal expansion of the isoperimetric quotient of a truncated cuboctahedron (great rhombicuboctahedron).

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A381690 Decimal expansion of the isoperimetric quotient of a snub cube (snub cuboctahedron).

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A381691 Decimal expansion of the isoperimetric quotient of an icosidodecahedron.

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