A381684 -id:A381684 - cp's OEIS frontend

A381694 Decimal expansion of the isoperimetric quotient of a (small) rhombicosidodecahedron.

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

Offset: 0

Paolo Xausa, Mar 08 2025

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

			0.93899527411045014134237823698302012883610912007046...

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

Cf. A344149 (surface area), A185093 (volume).

Cf. A000796, A002163, A002194, A381684.

First[RealDigits[4*Pi*(60 + 29*Sqrt[5])^2/(30 + Sqrt[75] + 3*Sqrt[25 + Sqrt[500]])^3, 10, 100]]

Equals 36*Pi*A185093^2/(A344149^3).

Equals 4*Pi*(60 + 29*sqrt(5))^2/((30 + 5*sqrt(3) + 3*sqrt(25 + 10*sqrt(5)))^3) = 4*A000796*(60 + 29*A002163)^2/((30 + 5*A002194 + 3*sqrt(25 + 10*A002163))^3).

A381695 Decimal expansion of the isoperimetric quotient of a truncated icosidodecahedron (great rhombicosidodecahedron).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 08 2025

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

			0.9135559084097273251197488307206577890586199166868...

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

Cf. A377796 (surface area), A377797 (volume).

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

First[RealDigits[Pi/30*(861 + 380*Sqrt[5])/(1 + Sqrt[3] + Sqrt[5 + Sqrt[20]])^3, 10, 100]]

Equals 36*Pi*A377797^2/(A377796^3).

Equals (Pi/30)*(861 + 380*sqrt(5))/((1 + sqrt(3) + sqrt(5 + 2*sqrt(5)))^3) = (A000796/30)*(861 + 380*A002163)/((1 + A002194 + sqrt(5 + A010476))^3).

A381696 Decimal expansion of the isoperimetric quotient of a snub dodecahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 10 2025

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

The snub dodecahedron is the Archimedean solid with the highest isoperimetric quotient.

			0.94699904523421562618454412877087470550479673815077...

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

Cf. A377804 (surface area), A377805 (volume).

Cf. A000796, A002163, A002194, A381684.

First[RealDigits[Pi/(20*Sqrt[3] + 3*Sqrt[25 + Sqrt[500]])^3*Root[#^6 - 52845*#^5 + 96583500*#^4 + 22087761875*#^3 + 4747626384375*#^2 - 5059009765293750*# + 187445810737515625 &, 4], 10, 100]]

Equals 36*Pi*A377805^2/(A377804^3).

Equals Pi/((20*sqrt(3) + 3*sqrt(25 + 10*sqrt(5)))^3)*r = A000796/((20*A002194 + 3*sqrt(25 + 10*A002163))^3)*r, where r is the largest real root of x^6 - 52845*x^5 + 96583500*x^4 + 22087761875*x^3 + 4747626384375*x^2 - 5059009765293750*x + 187445810737515625.

A382002 Decimal expansion of the isoperimetric quotient of a triakis tetrahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 16 2025

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

			0.64583578984055654756565980578430049996817368590574...

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

Cf. A378204 (surface area), A378205 (volume).

Cf. A000796, A010468, A381684.

First[RealDigits[15/22*Pi/Sqrt[11], 10, 100]]

Equals 36*Pi*A378205^2/(A378204^3).

Equals (15/22)*(Pi/sqrt(11)) = (15/22)*(A000796/A010468).

A382003 Decimal expansion of the isoperimetric quotient of a (small) triakis octahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 17 2025

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

			0.7900283767370127247375294315310284623115183154...

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

Cf. A378351 (surface area), A378352 (volume).

Cf. A000796, A002193, A381684.

First[RealDigits[Pi/51*Sqrt[(1399 + 988*Sqrt[2])/17], 10, 100]]

Equals 36*Pi*A378352^2/(A378351^3).

Equals (Pi/51)*sqrt((1399 + 988*sqrt(2))/17) = (A000796/51)*sqrt((1399 + 988*A002193)/17).

A382004 Decimal expansion of the isoperimetric quotient of a tetrakis hexahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 17 2025

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

			0.84297776772488716717876495718458737593598110244806...

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

Cf. A378388 (surface area), A374359 (volume - 1).

Cf. A000796, A002163, A381684.

First[RealDigits[3*Pi/5/Sqrt[5], 10, 100]]

Equals 36*Pi*(A374359 + 1)^2/(A378388^3).

Equals 3*Pi/(5*sqrt(5)) = (3/5)*A000796/A002163.

A382005 Decimal expansion of the isoperimetric quotient of a deltoidal icositetrahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 17 2025

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

			0.8697742819100637602738942626812998578199050663867...

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

Cf. A378390 (surface area), A378391 (volume).

Cf. A000796, A002193, A381684.

First[RealDigits[Pi/51*Sqrt[(3407 + 2384*Sqrt[2])/34], 10, 100]]

Equals 36*Pi*A378391^2/(A378390^3).

Equals (Pi/51)*sqrt((3407 + 2384*sqrt(2))/34) = (A000796/51)*sqrt((3407 + 2384*A002193)/34).

A382006 Decimal expansion of the isoperimetric quotient of a disdyakis dodecahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 18 2025

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

			0.91006563880803117059123808570537149844558354540595...

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

Cf. A378712 (surface area), A378713 (volume).

Cf. A000796, A002193, A381684.

First[RealDigits[Pi*7/97*Sqrt[(1709 + 1002*Sqrt[2])/194], 10, 100]]

Equals 36*Pi*A378713^2/(A378712^3).

Equals (Pi*7/97)*sqrt((1709 + 1002*sqrt(2))/194) = (A000796*7/97)*sqrt((1709 + 1002*A002193)/194).

A382007 Decimal expansion of the isoperimetric quotient of a pentagonal icositetrahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 19 2025

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

			0.87262832912869975551349997446851467573301874598462...

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

Cf. A378823 (surface area), A378824 (volume).

Cf. A000796, A381684.

First[RealDigits[Pi*Root[1936363968*#^6 - 149531184*#^4 + 10260*#^2 - 1 &, 2], 10, 100]]

Equals 36*Pi*A378824^2/(A378823^3).

Equals Pi*r = A000796*r, where r is the largest real root of 1936363968*x^6 - 149531184*x^4 + 10260*x^2 - 1.

A382008 Decimal expansion of the isoperimetric quotient of a rhombic triacontahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 20 2025

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

			0.88720000254802085800544409398426003785738986572116...

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

Cf. A344171 (surface area), A344172 (volume).

Cf. A000796, A098317, A381684.

First[RealDigits[Pi*(2 + Sqrt[5])/15, 10, 100]]

Equals 36*Pi*A344172^2/(A344171^3).

Equals Pi*(2 + sqrt(5))/15 = A000796*A098317/15.

cp's OEIS Frontend

A381694 Decimal expansion of the isoperimetric quotient of a (small) rhombicosidodecahedron.

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A381695 Decimal expansion of the isoperimetric quotient of a truncated icosidodecahedron (great rhombicosidodecahedron).

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A381696 Decimal expansion of the isoperimetric quotient of a snub dodecahedron.

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A382002 Decimal expansion of the isoperimetric quotient of a triakis tetrahedron.

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A382003 Decimal expansion of the isoperimetric quotient of a (small) triakis octahedron.

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A382004 Decimal expansion of the isoperimetric quotient of a tetrakis hexahedron.

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A382005 Decimal expansion of the isoperimetric quotient of a deltoidal icositetrahedron.

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A382006 Decimal expansion of the isoperimetric quotient of a disdyakis dodecahedron.

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