A105199 -id:A105199 - cp's OEIS frontend

A195710 Decimal expansion of arccos(-sqrt(2/5)).

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

Offset: 1

Clark Kimberling, Sep 23 2011

			arccos(-sqrt(2/5)) = 2.25551552979717...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A195708, A195701.

[Arccos(-Sqrt(2/5))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/5]; s = Sqrt[2/5];
N[ArcSin[r], 100]
RealDigits[%]  (* A073000 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A105199 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A188595 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A137218 *)
N[ArcSin[s], 100]
RealDigits[%]  (* A195701 *)
N[ArcCos[s], 100]
RealDigits[%]  (* A195708 *)
N[ArcTan[s], 100]
RealDigits[%]  (* A195709 *)
N[ArcCos[-s], 100]
RealDigits[%]  (* A195710 *)
RealDigits[ArcCos[-Sqrt[(2/5)]],10,120][[1]] (* Harvey P. Dale, Apr 06 2023 *)

acos(-sqrt(2/5)) \\ G. C. Greubel, Nov 18 2017

Equals Pi - arcsin(sqrt(3/5)) = Pi - arctan(sqrt(3/2)). - Amiram Eldar, Jul 08 2023

A197376 Decimal expansion of least x>0 having sin(x)=(sin (x/2))^2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 14 2011

The Mathematica program includes a graph. See A197133 for a guide to least x>0 satisfying sin(b*x)=(sin(c*x))^2 for selected b and c.

			2.2142974355881810060341309203570740...

Index entries for transcendental numbers.

Cf. A197133.

b = 1; c = 1/2; f[x_] := Sin[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 2, 2.5}, WorkingPrecision -> 200]
RealDigits[t] (*  *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 2.5}]
RealDigits[ 2*ArcCos[ 1/Sqrt[5] ], 10, 99] // First (* Jean-François Alcover, Feb 19 2013 *)

Equals 2*A105199. - R. J. Mathar, Aug 27 2024

A383859 Central angle of the solution of the Tammes problem for 7 points on the sphere (in radians).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, May 12 2025

			1.3590798976326601418850028816473327537..

Laslo Hars, Numerical solutions of the Tammes problem for up to 60 points, Nov 2020, N=7.
K. Schutte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand eins Platz?, Math. Ann. 123 (1951) 96-124.
Wikipedia, Tammes problem

Cf. A019819, A019669 (N=6), A381756 (N=8), A137914 (N=9), A340918 (N=10), A105199 (N=11 and N=12). A217695 (N=13), A383860 (N=14), A383861 (N=24).

cos(4*Pi/9) ; %/(1-%) ; arccos(%) ; evalf(%,120) ;

cos( this ) = cos phi/(1- cos phi) where cos(phi)=A019819.

A383860 Central angle of the solution of the Tammes problem for 14 points on the sphere (in radians).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, May 12 2025

			0.971634742886224075941694947628...

Laslo Hars, Numerical solutions of the Tammes problem for up to 60 points, Nov 2020, N=14.
O. R. Musin and A. S. Tarasov, The Tammes problem for N=14, Exp. Math. 24 (2015) 460-468.
Wikipedia, Tammes problem

Cf. A019669 (N=6), A383859 (N=7), A381756 (N=8), A137914 (N=9), A340918 (N=10), A105199 (N=11 and N=12). A217695 (N=13), A383861 (N=24).

Digits := 120 ;
g := proc(c,x)
    2*arccot(c*tan(x/2)) ;
end proc:
f := proc(x)
    local c,x1,x2,x3,x4,x5 ;
    c := cos(x)/(1-cos(x)) ;
    x1 := Pi-x ;
    x2 := g(c,x1) ;
    x3 := 2*Pi-2*x-x2 ;
    x4 := g(c,x3) ;
    x5 := 2*Pi-x-2*x4 ;
    2*Pi-2*x-x3-g(c,x5) ;
end proc:
x := 1.2 ; y := 1.21 ;
for i from 1 to 500 do
    z := (x+y)/2 ;
    if f(z) > 0. then
        x := z ;
    else
        y := z ;
    end if;
    cos(z)/(1-cos(z)) ;
    if modp(i,20) =0 then
        arccos(%) ; evalf(%,120) ; print(%) ;
    end if;
    if x > y then
        break ;
    end if;
end do:

A383861 Central angle of the solution of the Tammes problem for 24 points on the sphere (in radians).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, May 12 2025

			0.762547738750982556743106092114...

Laslo Hars, Numerical solutions of the Tammes problem for up to 60 points, Nov 2020, N=24.
R. M. Robinson, Arrangements of 24 points on the sphere, Math. Ann. 144 (1961) 17-48.
Wikipedia, Tammes problem

Cf. A058265, A019669 (N=6), A383859 (N=7), A381756 (N=8), A137914 (N=9), A340918 (N=10), A105199 (N=11 and N=12). A217695 (N=13), A383860 (N=14).

t := (1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3))/3 ; arccos((t-1)/(3-t)) ; evalf(%,120);

cos( this ) = (t-1)/(3-t) where t=A058265.

A387191 Decimal expansion of the second largest dihedral angle, in radians, in an elongated pentagonal rotunda (Johnson solid J_21).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 22 2025

This is the dihedral angle between a square face and a pentagonal face.

Also one of the dihedral angles in Johnson solids J_40-J_43, J_72-J_75, J_77-J_79 and J_82.

			2.677945044588987122248387151818288482168632345...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Elongated pentagonal rotunda.

Cf. other J_21 dihedral angles: A019669, A228824, A344075, A386530.
Cf. A384213 (J_21 volume), A179637 (J_21 surface area - 10).
Cf. A010476, A105199.

First[RealDigits[Pi/2 + ArcTan[2], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J21", "DihedralAngles"]], 2], 10, 100]]

Equals Pi/2 + arctan(2) = A019669 + A105199.

Equals arccos(-2*sqrt(5)/5) = arccos(-A010476/5).

A386732 Decimal expansion of Integral_{x>=2} 1/(x^12-1) dx.

Original entry on oeis.org

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

Offset: 0

Jason Bard, Jul 31 2025

			0.000044394388389732931619793708861045902941185047688518...

Jason Bard, Table of n, a(n) for n = 0..1003
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21.

Cf. A016644, A105199, A123868, A304656.

Join[{0, 0, 0, 0}, RealDigits[1/72 (-4 (3 + Sqrt[3]) Pi + 3 (4 ArcTan[2] + 2 Sqrt[3] ArcTan[5/Sqrt[3]] + 2 ArcTan[4 - Sqrt[3]] + 2 ArcTan[4 + Sqrt[3]] + Log[21] - Sqrt[3] Log[5 - 2 Sqrt[3]] + Sqrt[3] Log[5 + 2 Sqrt[3]])), 10, 100][[1]]]
(* or *)
Join[{0, 0, 0, 0}, RealDigits[Integrate[1/(x^12 - 1), {x, 2, Infinity}], 10, 100][[1]]]
(* or *)
Join[{0, 0, 0, 0}, RealDigits[1/22528*Hypergeometric2F1[11/12, 1, 23/12, 1/4096], 10, 100][[1]]]

Equals (1/22528) * hypergeometric(11/12, 1; 23/12; 1/4096).

Equals (-6*Pi - 4*sqrt(3)*Pi + 12*arctan(2) - 3*arctan(12/5) + 6*sqrt(3) * arctan(5/sqrt(3)) + 6*sqrt(3) * arctanh((2*sqrt(3))/5) + log(9261))/72.

A386853 Decimal expansion of the dihedral angle, in radians, between the 10-gonal face and a triangular face in a pentagonal rotunda (Johnson solid J_6).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 06 2025

			1.38208579601133454945018729145714326976181383401...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Pentagonal rotunda.
Index entries for transcendental numbers.

Cf. A179593 (volume), A179637 (surface area).
Cf. other J_6 dihedral angles: A105199, A344075.
Cf. A010476, A386852.

First[RealDigits[ArcCos[Sqrt[(5 - Sqrt[20])/15]], 10, 100]] (* or *)
First[RealDigits[RankedMin[Union[PolyhedronData["J6", "DihedralAngles"]], 2], 10, 100]]

acos(sqrt((5 - 2*sqrt(5))/15)) \\ Charles R Greathouse IV, Aug 19 2025

Equals arccos(sqrt((5 - 2*sqrt(5))/15)) = arccos(sqrt((5 - A010476)/15)).

A387189 Decimal expansion of the smallest dihedral angle, in radians, in a pentagonal bipyramid (Johnson solid J_13).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 21 2025

This is the dihedral angle between triangular faces at the edge where the two pyramidal parts of the solid meet.

Also the dihedral angle between triangular faces in a pentagonal orthobicupola (Johnson solid J_30).

			1.3047162795687363719907812632876451487306158399...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Pentagonal bipyramid.
Wikipedia, Pentagonal orthobicupola.

Cf. A236367 (J_13 smallest dihedral angle).
Cf. other J_30 dihedral angles: A105199, A377995, A377996.
Cf. A179641 (J_13 volume), A120011 (J_13 surface area, divided by 10).
Cf. A384624 (J_30 volume), A384625 (J_30 surface area).
Cf. A010532, A386852.

First[RealDigits[ArcCos[(Sqrt[80] - 5)/15], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J13", "DihedralAngles"]], 10, 100]]

Equals arccos((4*sqrt(5) - 5)/15) = arccos((A010532 - 5)/15).

Equals 2*A386852.

A366313 a(n) = Product_{k=0..2*n} (n^2 + k^2).

Original entry on oeis.org

0, 10, 41600, 805545000, 48248012800000, 6993773647152500000, 2092947132921735168000000, 1157435764584534017163490000000, 1090228457517544945858327347200000000, 1643200095810939801357184785754425000000000

Offset: 0

Vaclav Kotesovec, Oct 06 2023

nonn

Mathematics StackExchange, A limit with infinite product, 2013.

Cf. A272244, A105199.

Table[Product[k^2 + n^2, {k, 0, 2*n}], {n, 0, 10}]
Table[n^2*Pochhammer[1 - I*n, 2*n]*Pochhammer[1 + I*n, 2*n], {n, 0, 10}]

a(n) = prod(k=0, 2*n, n^2 + k^2); \\ Michel Marcus, Oct 06 2023

a(n) = n * sinh(n*Pi) * Gamma(1 + (2-i)*n) * Gamma(1 + (2+i)*n)/Pi, where i is the imaginary unit.

a(n) ~ 5^(2*n + 1/2) * exp(2*n*(arctan(2) - 2)) * n^(4*n+2).

cp's OEIS Frontend

A195710 Decimal expansion of arccos(-sqrt(2/5)).

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A197376 Decimal expansion of least x>0 having sin(x)=(sin (x/2))^2.

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A383859 Central angle of the solution of the Tammes problem for 7 points on the sphere (in radians).

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Maple

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A383860 Central angle of the solution of the Tammes problem for 14 points on the sphere (in radians).

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Maple

A383861 Central angle of the solution of the Tammes problem for 24 points on the sphere (in radians).

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Maple

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A387191 Decimal expansion of the second largest dihedral angle, in radians, in an elongated pentagonal rotunda (Johnson solid J_21).

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Mathematica

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A386732 Decimal expansion of Integral_{x>=2} 1/(x^12-1) dx.

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A386853 Decimal expansion of the dihedral angle, in radians, between the 10-gonal face and a triangular face in a pentagonal rotunda (Johnson solid J_6).

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