A188582 -id:A188582 - cp's OEIS frontend

A378352 Decimal expansion of the volume of a (small) triakis octahedron with unit shorter edge length.

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

Offset: 1

Paolo Xausa, Nov 23 2024

The (small) triakis octahedron is the dual polyhedron of the truncated cube.

			2.9142135623730950488016887242096980785696718753769...

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

Cf. A378351 (surface area), A378353 (inradius), A201488 (midradius), A378354 (dihedral angle).
Cf. A377299 (volume of a truncated cube with unit edge).
Cf. A156035.
Essentially the same as A002193 and A188582.

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

Equals sqrt(2) + 3/2 = A002193 + 3/2.

Equals A156035/2. - Hugo Pfoertner, Nov 24 2024

A343057 Decimal expansion of tan(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.098491403357164253077197...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8.

Cf. A343055 (sin(Pi/32)), A343056 (cos(Pi/32)).

tan(Pi/m): A002194 (m=3), A019934 (m=5), A020760 (m=6), A343058 (m=7), A188582 (m=8), A019918 (m=9), A019916 (m=10), A019913 (m=12), A343059 (m=14), A019910 (m=15), A343060 (m=16), A343061 (m=17), A019908 (m=18), A019907 (m=20), A343062 (m=24), A019904 (m=30), A343057 (m=32), A019903 (m=36).

R:= RealField(125); Tan(Pi(R)/32); // G. C. Greubel, Sep 30 2022

RealDigits[Tan[Pi/32], 10, 120, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/32)
```

sqrt((2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))))

numerical_approx(tan(pi/32), digits=125) # G. C. Greubel, Sep 30 2022

Equals sqrt( (2-sqrt(2+sqrt(2+sqrt(2))))/(2+sqrt(2+sqrt(2+sqrt(2)))) ).

A270396 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/Fibonacci(k+1).

Original entry on oeis.org

3, 7, 36, 1040, 1378784, 2783678150237, 20812561896916543523976387, 398006071848302987834283599453836703483929049938762, 105246367677020752496441044566935490666701848819994695873528056638957197400663802988967689301303582936

Offset: 1

Clark Kimberling, Mar 22 2016

Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ..., the r-Egyptian fraction for x.

See A269993 for a guide to related sequences.

			sqrt(2) - 1 = 1/3 + 1/(2*7) + 1/(3*36) + 1/(5*1040) + ...

Clark Kimberling, Table of n, a(n) for n = 1..11
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A000045, A188582.

r[k_] := 1/Fibonacci[k+1]; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[2] - 1; Table[n[x, k], {k, 1, z}]

r(k) = 1/fibonacci(k+1);
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016

A270548 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/(2k-1).

Original entry on oeis.org

3, 5, 15, 163, 29203, 1370794960, 5693192315226228214, 247405800822801380465687897681838336769, 267682228701778523205506744045084667800917057557706608910309126004853790212423

Offset: 1

Clark Kimberling, Apr 02 2016

Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.

See A269993 for a guide to related sequences.

			sqrt(2) - 1 = 1/(1*3) + 1/(3*5) + 1/(5*15) + 1/(7*163) + ...

Clark Kimberling, Table of n, a(n) for n = 1..12
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A005408, A188582.

r[k_] := 1/(2k-1); f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt(2) - 1; Table[n[x, k], {k, 1, z}]

r(k) = 1/(2*k-1);
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Apr 03 2016

A297701 Decimal expansion of 1 + sqrt(2) + sqrt(3).

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Jan 03 2018

This is an algebraic integer of degree 4, with minimal polynomial x^4 - 4*x^3 - 4*x^2 + 16*x - 8.

			  1.0000000000000000000000000000...
+ 1.4142135623730950488016887242...
+ 1.7320508075688772935274463415...
= 4.1462643699419723423291350657...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 4

Essentially the same as A135611. Cf. A002193, A002194, A014176, A165663, A188582.

SetDefaultRealField(RealField(100)); 1 + Sqrt(2) + Sqrt(3); // G. C. Greubel, Nov 20 2018

RealDigits[1 + Sqrt[2] + Sqrt[3], 10, 100][[1]]

1+sqrt(2)+sqrt(3) \\ Felix Fröhlich, Jan 06 2018

numerical_approx(1+sqrt(2)+sqrt(3), digits=100) # G. C. Greubel, Nov 20 2018

1 + sqrt(2) + sqrt(3) = 1 + sqrt(5 + 2 sqrt(6)).

1 + A002193 + A002194 = A014176 + A002194 = A165663 + A188582. - Felix Fröhlich, Jan 06 2018

Terms a(52) onward corrected by G. C. Greubel, Nov 20 2018

A270519 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/k!.

Original entry on oeis.org

3, 7, 18, 217, 21586, 132830816, 8232750479147118, 8738244742575919521189548340591, 28575128242342620144630216663972970082807062570299713849045286

Offset: 1

Clark Kimberling, Mar 30 2016

Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.

See A269993 for a guide to related sequences.

			sqrt(2) - 1 = 1/(1*3) + 1/(2*7) + 1/(6*18) + 1/(24*217) + ...

Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A000142, A188582.

r[k_] := 1/k!; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[2] - 1; Table[n[x, k], {k, 1, z}]

r(k) = 1/k!;
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 31 2016

A319905 Decimal expansion of 4*(sqrt(2) - 1)/3.

Original entry on oeis.org

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

Offset: 0

Franck Maminirina Ramaharo, Oct 01 2018

A 90-degree unit-circular arc in the first quadrant can be approximated by a cubic Bézier curve. In this case, L = 4*(sqrt(2) - 1)/3 is the unit tangent vector scaling factor that minimizes the distance between the curve and the unit circle segment, provided its endpoints and midpoint are interpolated.
Riškus referred to this constant as "magic number".
The Bézier curve with control points {(1,0), (1,L), (L,1), (0,1)} has a minimum distance to the origin of 1 (at t in {0, 1/2, 1}), and it has a maximum distance to the origin of (1/3)*sqrt(71/6-2*sqrt(2)) = 1.00027253... at t in {(3 - sqrt(3))/6,(3 + sqrt(3))/6}. - Peter Kagey, Feb 21 2025

			0.552284749830793398402251632279597438092895833835930...

Tor Dokken, Morten Dæhlen, Tom Lyche and Knut Mørken, Good approximation of circles by curvature-continuous Bézier curves, Computer Aided Geometric Design Vol. 7 (1990), 33-41.
Aleksas Riškus, Approximation of a cubic Bézier curve by circular arcs and vice versa, Information Technology And Control Vol. 35 (2006), 371-378.
Adam G. Stanislav, Drawing a circle with Bézier Curves
Wikipedia, Bézier curve
Wikipedia, Composite Bézier curve

Cf. A002193, A156309, A188582, A268683.

Maple
```
Digits:=1000; evalf(4*(sqrt(2) - 1)/3);
```

RealDigits[4*(Sqrt[2] - 1)/3, 10, 100][[1]]

PARI
```
4*(sqrt(2) - 1)/3
```

Equals (4/3)*tan(Pi/8).

Irrational number represented by the periodic continued fraction [0; [1, 1, 4, 3]]; positive real root of 9*x^2 + 24*x - 16. - Peter Luschny, Oct 04 2018

A367480 Decimal expansion of the radius of a common circle surrounded by seven tangent unit circles.

Original entry on oeis.org

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

Offset: 1

Thomas Otten, Dec 23 2023

The radius of a common circle surrounded by n tangent unit circles (n > 2) is r = 1/sin(Pi/n) - 1.

n=7 is the smallest number for which the radius cannot be expressed using square roots, since the regular heptagon formed by the centers of the tangent circles is non-constructible (see A246724, A188582, and A121570 for n=3, 4, 5).

			1.3047648709624865052...

Andrew M. Gleason, Angle Trisection, the Heptagon, and the Triskaidecagon, The American Mathematical Monthly 95, no. 3 (March 1988), pp. 185-194.
Index entries for algebraic numbers, degree 6.

Cf. A121598.

Cf. A121570, A188582, A246724.

RealDigits[Csc[Pi/7] - 1, 10, 120][[1]] (* Amiram Eldar, Dec 28 2023 *)

PARI
```
1/sin(Pi/7) - 1
```

Equals 1 / sin(Pi/7) - 1.
Equals A121598 - 1.
Largest of the 6 real-valued roots of 7*x^6+ 42*x^5 +49*x^4 -84*x^3 -119*x^2 +42*x-1=0. - R. J. Mathar, Aug 29 2025

More digits from Jon E. Schoenfield, Dec 24 2023

Comments edited by Michal Paulovic, Dec 26 2023

cp's OEIS Frontend

A378352 Decimal expansion of the volume of a (small) triakis octahedron with unit shorter edge length.

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A343057 Decimal expansion of tan(Pi/32).

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A270396 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/Fibonacci(k+1).

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A270548 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/(2k-1).

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A297701 Decimal expansion of 1 + sqrt(2) + sqrt(3).

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A270519 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/k!.

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A319905 Decimal expansion of 4*(sqrt(2) - 1)/3.

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