A002194 -id:A002194 - cp's OEIS frontend

A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a regular dodecahedron to the volume of the circumscribed sphere (which has circumradius a*(sqrt(3) + sqrt(15))/4 = a*(A002194 + A010472)/4, where a is the dodecahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165954. A063723 shows the order of these by size.

			0.6649088942053266431144284467086337161648765805556919381057592605722964718...

Eric Weisstein's World of Mathematics, Dodecahedron.
Index entries for transcendental numbers.

Cf. A000796, A002194, A010472, A165922, A049541, A165952, A165954, A063723, A002163, A020760, A010527.

RealDigits[(5*Sqrt[3]+Sqrt[15])/(6*Pi),10,120][[1]] (* Harvey P. Dale, Feb 16 2018 *)

PARI
```
(5*sqrt(3)+sqrt(15))/(6*Pi)
```

Equals (5*A002194 + A010472)/(6*A000796).
Equals (5*A002194 + A010472)*A049541/6.
Equals (10*A010527 + A010472)*A049541/6.
Equals (5 + sqrt(5))/(2*Pi*sqrt(3)).
Equals (5 + A002163)*A049541*A020760/2.

A171537 Decimal expansion of sqrt(3/7).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient = <2 3/2 ; -2 -1/2 | 7/2 -5/2>.

			sqrt(3/7) = 0.6546536707079771437982924562...

Wikipedia, Table of Clebsch-Gordan coefficients

RealDigits[N[Sqrt[3/7],300]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

equals A002194/A010465 = 3/A010477.

A176053 Decimal expansion of (3+2*sqrt(3))/3.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (3+2*sqrt(3))/3 is A010696.
a(n) = A020832(n-1) for n > 1; a(1) = 2.
This equals the ratio of the radius of the outer Soddy circle and the common radius of the three kissing circles. See A343235, also for links. - Wolfdieter Lang, Apr 19 2021
Previous comment is, together with A246724, the answer to the 1st problem proposed during the 4th Canadian Mathematical Olympiad in 1972. - Bernard Schott, Mar 20 2022

			2.15470053837925152901...

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993 - Canadian Mathematical Society & Société Mathématique du Canada, Problem 1, 1972, page 37, 1993.

Paolo Xausa, Table of n, a(n) for n = 1..10000
The IMO Compendium, Problem 1, 4th Canadian Mathematical Olympiad, 1972.
Michael Penn, An inscribed tower of squares, YouTube video, 2020.
Bernard Schott, Illustration of the Soddy circles.
Index to sequences related to Olympiads.

Cf. A002194 (sqrt(3)), A020832 (1/sqrt(75)), A010696 (repeat 2, 6).

Cf. A246724, A343235.

RealDigits[1+2/3Sqrt[3],10,100][[1]] (* Paolo Xausa, Aug 10 2023 *)

Equals 2 + A246724.

A176322 Decimal expansion of sqrt(1365).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of sqrt(1365) is (repeat 1, 17, 2, 17, 1, 72) preceded by 36.

			36.94590640382233199163...

G. C. Greubel, Table of n, a(n) for n = 2..1001

Cf. A002194 (sqrt(3)), A002163 (sqrt(5)), A010465 (sqrt(7)), A010470 (sqrt(13)).

Cf. A176321 ((35+sqrt(1365))/14).

SetDefaultRealField(RealField(120)); Sqrt(1365); // G. C. Greubel, Nov 26 2019

evalf( sqrt(1365), 120); # G. C. Greubel, Nov 26 2019

RealDigits[Sqrt[1365],10,120][[1]] (* Harvey P. Dale, Oct 09 2017 *)

default(realprecision, 120); sqrt(1365) \\ G. C. Greubel, Nov 26 2019

numerical_approx(sqrt(1365), digits=120) # G. C. Greubel, Nov 26 2019

Equals sqrt(3)*sqrt(5)*sqrt(7)*sqrt(13).

A184797 Numbers k such that floor(k*sqrt(3)) is prime.

Original entry on oeis.org

2, 3, 8, 10, 11, 17, 18, 24, 25, 31, 39, 41, 46, 48, 60, 62, 63, 76, 91, 100, 105, 112, 114, 115, 122, 129, 135, 138, 145, 152, 157, 160, 180, 181, 195, 202, 204, 212, 219, 225, 232, 242, 249, 250, 254, 256, 264, 270, 277, 284, 294, 301, 302, 316, 322, 329, 330, 339, 346, 347, 351, 354, 374, 381, 382, 389, 391, 399, 405, 420, 427, 429, 434, 444, 478, 479, 493, 495, 496, 509, 510, 524, 526, 531, 541, 547, 561, 568, 576, 583, 585, 590, 600

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

			See A184796.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002194, A022838, A184796, A184798.

(See A184796.)
Select[Range[600],PrimeQ[Floor[# Sqrt[3]]]&] (* Harvey P. Dale, May 21 2017 *)

is(n)=isprime(sqrtint(3*n^2)) \\ Charles R Greathouse IV, May 22 2017

A241149 Decimal expansion of sqrt(2) + sqrt(3) + sqrt(5).

Original entry on oeis.org

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

Offset: 1

K. D. Bajpai, Apr 16 2014

This is an algebraic integer, with its minimal polynomial being x^8 - 40x^6 + 352x^4 - 960x^2 + 576. - Alonso del Arte, Apr 17 2014

			5.382332347441762038738308734446846680953095488798854425503383962...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Penn, When showing square root 2 is irrational is too easy, YouTube video, 2022.

Cf. A002163 (decimal expansion: sqrt(5)).
Cf. A002193 (decimal expansion: sqrt(2)).
Cf. A002194 (decimal expansion: sqrt(3)).

Sqrt(2) + Sqrt(3) + Sqrt(5); // G. C. Greubel, Jul 27 2018

evalf(add(sqrt(ithprime(i)), i=1..3), 121);  # Alois P. Heinz, Jun 13 2022

RealDigits[Sqrt[2] + Sqrt[3] + Sqrt[5], 10, 200]

sqrt(2) + sqrt(3) + sqrt(5) \\ G. C. Greubel, Jul 27 2018

A265294 Decimal expansion of Sum_{n>=1} (x - c(2n-1)), where c = convergents to (x = sqrt(3)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 07 2015

			sum = 0.8025830908035148343778741812630...

Cf. A002194, A002530, A002531, A265295, A265296, A265288 (guide).

x := sqrt(3) - 2:
evalf(2*sqrt(3)*add( x^(n*(n+1)/2)/(x^n - 1), n = 1..18), 100); # Peter Bala, Aug 24 2022

x = Sqrt[3]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265294 *)
RealDigits[s2, 10, 120][[1]]  (* A265295 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)

From Peter Bala, Aug 24 2022: (Start)
Equals 2*sqrt(3)*Sum_{n >= 1} 1/( 1 + (2+sqrt(3))^(2*n-1) ).
A more rapidly converging series for the constant is 2*sqrt(3)*Sum_{n >= 1} x^(n*(n+1)/2)/(x^n - 1), where x = sqrt(3) - 2. See A001227. (End)

A265295 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(3).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 07 2015

			sum = 0.28728008008348839351145153987668331682390...

Cf. A002194, A002530, A002531, A265294, A265296, A265288 (guide).

x := 7 - 4*sqrt(3):
evalf(2*sqrt(3)*add( x^(n^2)*(1 + x^n)/(1 - x^n), n = 1..10), 100); # Peter Bala, Aug 24 2022

x = Sqrt[3]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265294 *)
RealDigits[s2, 10, 120][[1]]  (* A265295 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)

Equals 2*sqrt(3)*Sum_{n >= 1} x^(n^2)*(1 + x^n)/(1 - x^n), where x = 7 - 4*sqrt(3). - Peter Bala, Aug 24 2022

A265296 Decimal expansion of Sum_{n >= 1} (c(2n) - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 07 2015

			sum = 1.0898631708870032278893257211397258128825141977596999...

Cf. A002194, A002530, A002531, A265294, A265295, A265288 (guide).

x := 2 - sqrt(3):
evalf(2*sqrt(3)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..13), 100); # Peter Bala, Aug 24 2022

x = Sqrt[3]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265294 *)
RealDigits[s2, 10, 120][[1]]  (* A265295 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)

Equals 2*sqrt(3)*Sum_{n >= 1} x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), where x = 2 - sqrt(3). - Peter Bala, Aug 24 2022

A272535 Decimal expansion of the edge length of a regular 16-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 02 2016

Like all m-gons with m equal to a power of 2 (see A003401 and A000079), this is a constructible number.

			0.390180644032256535696569736954044481855383235503909615509004...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 8.

Cf. A000079, A003401.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A228787 (17), A272536 (20).

RealDigits[N[2Sin[Pi/16], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/16)
```

Equals 2*sin(Pi/m) for m=16, 2*A232738. Equals also sqrt(2-sqrt(2+sqrt(2))).

cp's OEIS Frontend

A165953 Decimal expansion of (5*sqrt(3) + sqrt(15))/(6*Pi).

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A171537 Decimal expansion of sqrt(3/7).

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A176053 Decimal expansion of (3+2*sqrt(3))/3.

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A176322 Decimal expansion of sqrt(1365).

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A184797 Numbers k such that floor(k*sqrt(3)) is prime.

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A241149 Decimal expansion of sqrt(2) + sqrt(3) + sqrt(5).

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Maple

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A265294 Decimal expansion of Sum_{n>=1} (x - c(2n-1)), where c = convergents to (x = sqrt(3)).

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A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

A265296 Decimal expansion of Sum_{n >= 1} (c(2n) - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(3).