A377697 -id:A377697 - cp's OEIS frontend

A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

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

Offset: 0

N. J. A. Sloane

Decimal expansion of cos(2*Pi/5) (angle of 72 degrees).
Also the imaginary part of i^(1/5). - Stanislav Sykora, Apr 25 2012
One of the two roots of 4x^2 + 2x - 1 (the other is the sine of 54 degrees times -1 = -A019863). - Alonso del Arte, Apr 25 2015
This is the height h of the isosceles triangle in a regular pentagon inscribed in a unit circle, formed by a diagonal as base and two adjacent radii. h = cos(2*Pi/5) = sin(Pi/10). - Wolfdieter Lang, Jan 08 2018
Quadratic number of denominator 2 and minimal polynomial 4x^2 + 2x - 1. - Charles R Greathouse IV, May 13 2019
Largest superstable width of the logistic map (see Finch). - Stefano Spezia, Nov 23 2024

			0.30901699437494742410229341718281905886015458990288143106772431135263...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.9 and 8.19, pp. 66, 535.

Zak Seidov, Table of n, a(n) for n = 0..999
Hideyuki Ohtsuka, Problem B-1237, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 4 (2018), p. 366; A Telescoping Product, Solution to Problem B-1237 by Steve Edwards, ibid., Vol. 57, No. 4 (2019), pp. 369-370.
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 2.

Cf. A001622, A019845, A019863, A055588, A094214, A134945, A187798, A341332, A377697.

RealDigits[Sin[18 Degree], 10, 108][[1]] (* Alonso del Arte, Apr 20 2015 *)

sin(Pi/10) \\ Charles R Greathouse IV, Feb 03 2015

polrootsreal(4*x^2 + 2*x - 1)[2] \\ Charles R Greathouse IV, Feb 03 2015

Equals (sqrt(5) - 1)/4 = (phi - 1)/2 = 1/(2*phi), with phi from A001622.
Equals 1/(1 + sqrt(5)). - Omar E. Pol, Nov 15 2007
Equals 1/A134945. - R. J. Mathar, Jan 17 2021
Equals 2*A019818*A019890. - R. J. Mathar, Jan 17 2021
Equals Product_{k>=1} 1 - 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - Amiram Eldar, Dec 02 2021
Equals Product_{k>=1} (1 - 1/A055588(k)). - Amiram Eldar, Nov 28 2024
Equals A094214/2 = 1-A187798 = A341332/Pi = (A377697-2)/3. - Hugo Pfoertner, Nov 28 2024
This^2 + A019881^2 = 1. - R. J. Mathar, Aug 31 2025

A377694 Decimal expansion of the surface area of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Nov 04 2024

			100.990760153101988544745948988636656554915090575...

Eric Weisstein's World of Mathematics, Truncated Dodecahedron.
Wikipedia, Truncated dodecahedron.

Cf. A377695 (volume), A377696 (circumradius), A377697 (midradius), A377698 (Dehn invariant, negated).
Cf. A131595 (analogous for a regular dodecahedron).
Cf. A002194, A010476.

First[RealDigits[5*(Sqrt[3] + 6*Sqrt[5 + Sqrt[20]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "SurfaceArea"], 10, 100]]

Equals 5*(sqrt(3) + 6*sqrt(5 + 2*sqrt(5))) = 5*(A002194 + 6*sqrt(5 + A010476)).

A377695 Decimal expansion of the volume of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 04 2024

			85.039664559370881554679651012654159610712109542...

Eric Weisstein's World of Mathematics, Truncated Dodecahedron.
Wikipedia, Truncated dodecahedron.

Cf. A377694 (surface area), A377696 (circumradius), A377697 (midradius), A377698 (Dehn invariant, negated).
Cf. A102769 (analogous for a regular dodecahedron).
Cf. A002163.

First[RealDigits[5/12*(99 + 47*Sqrt[5]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "Volume"], 10, 100]]

Equals (5/12)*(99 + 47*sqrt(5)) = (5/12)*(99 + 47*A002163).

A378976 Decimal expansion of the midradius of a triakis icosahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 14 2024

The triakis icosahedron is the dual polyhedron of the truncated dodecahedron.

			1.3944271909999158785636694674925104941762473438446...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triakis Icosahedron.
Wikipedia, Triakis icosahedron.

Cf. A378973 (surface area), A378974 (volume), A378975 (inradius), A378977 (dihedral angle).
Cf. A377697 (midradius of a truncated dodecahedron with unit edge length).
Cf. A002163, A010532, A249600.

First[RealDigits[1/2 + 2/Sqrt[5], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisIcosahedron", "Midradius"], 10, 100]]

Equals 1/2 + 2/sqrt(5) = 1/2 + 2/A002163.

Equals (A249600 + 13)/10 = (A010532 + 5)/10.

A385804 Decimal expansion of the volume of a triaugmented dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 09 2025

The triaugmented dodecahedron is Johnson solid J_61.

			8.56762745781210568076720062887114294145115942427...

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

Cf. A385805 (surface area).

Cf. A010499, A102769, A179552, A377697, A385695, A385802.

First[RealDigits[5/8*(7 + Sqrt[45]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J61", "Volume"], 10, 100]]

Equals (5/8)*(7 + 3*sqrt(5)) = (5/8)*(7 + A010499).
Equals A102769 + 3*A179552.
Equals the largest root of 16*x^2 - 140*x + 25.
Equals A377697^2. - Hugo Pfoertner, Jul 13 2025

A377696 Decimal expansion of the circumradius of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 04 2024

			2.9694490158633984670421666956925979636007477003...

Eric Weisstein's World of Mathematics, Truncated Dodecahedron.
Wikipedia, Truncated dodecahedron.

Cf. A377694 (surface area), A377695 (volume), A377697 (midradius), A377698 (Dehn invariant, negated).
Cf. A179296 (analogous for a regular dodecahedron).
Cf. A002163.

First[RealDigits[Sqrt[74 + 30*Sqrt[5]]/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "Circumradius"], 10, 100]]

Equals sqrt(74 + 30*sqrt(5))/4 = sqrt(74 + 30*A002163)/4.

cp's OEIS Frontend

A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

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A377694 Decimal expansion of the surface area of a truncated dodecahedron with unit edge length.

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A377695 Decimal expansion of the volume of a truncated dodecahedron with unit edge length.

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A378976 Decimal expansion of the midradius of a triakis icosahedron with unit shorter edge length.

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A385804 Decimal expansion of the volume of a triaugmented dodecahedron with unit edge.

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A377696 Decimal expansion of the circumradius of a truncated dodecahedron with unit edge length.

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