A002194 -id:A002194 - cp's OEIS frontend

A131595 Decimal expansion of 3(sqrt(25 + 10sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.

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

Offset: 2

Omar E. Pol, Aug 30 2007

Surface area of a regular dodecahedron: A = 3*(sqrt(25 + 10*sqrt(5)))* a^2, where 'a' is the edge.

			20.64572880706760307310814372866331519288849004012237995...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

G. C. Greubel, Table of n, a(n) for n = 2..10001
Eric Weisstein's World of Mathematics, Dodecahedron.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A102769, A001622 (phi), A182007 (associate of phi), A010527 (icosahedron/10), A010469 (octahedron), A002194 (tetrahedron). - Stanislav Sykora, Nov 30 2013

SetDefaultRealField(RealField(100)); 3*(Sqrt(25 + 10*Sqrt(5))); // G. C. Greubel, Nov 02 2018

evalf(3*(sqrt(25+10*sqrt(5))),130); # Muniru A Asiru, Nov 02 2018

RealDigits[3*Sqrt[25+10*Sqrt[5]],10,120][[1]] (* Harvey P. Dale, Jun 21 2011 *)

default(realprecision, 100); 3*(sqrt(25 + 10*sqrt(5))) \\ G. C. Greubel, Nov 02 2018

From Stanislav Sykora, Nov 30 2013: (Start)
Equals 15/tan(Pi/5).
Equals 15*phi/xi, where phi is the golden ratio (A001622) and xi its associate (A182007). (End)

More terms from Harvey P. Dale, Jun 21 2011

A005320 a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3.

Original entry on oeis.org

0, 3, 12, 45, 168, 627, 2340, 8733, 32592, 121635, 453948, 1694157, 6322680, 23596563, 88063572, 328657725, 1226567328, 4577611587, 17083879020, 63757904493, 237947738952, 888033051315, 3314184466308, 12368704813917, 46160634789360, 172273834343523, 642934702584732, 2399464975995405, 8954925201396888, 33420235829592147

Offset: 0

N. J. A. Sloane

For n > 1, a(n-1) is the determinant of the n X n band matrix which has {2,4,4,...,4,4,2} on the diagonal and a 1 on the entire super- and subdiagonal. This matrix appears when constructing a natural cubic spline interpolating n equally spaced data points. - g.degroot(AT)phys.uu.nl, Feb 14 2007
Integer values of x that make 9+3*x^2 a perfect square. - Lorenz H. Menke, Jr., Mar 26 2008
The intermediate convergents to 3^(1/2), beginning with 3/2, 12/7, 45/26, 168/97, comprise a strictly increasing sequence whose numerators are the terms of this sequence and denominators are A001075. - Clark Kimberling, Aug 27 2008
a(n) also give the altitude to the middle side of a Super-Heronian Triangle. - Johannes Boot, Oct 14 2010
a(n) gives values of y satisfying 3*x^2 - 4*y^2 = 12; corresponding x values are given by A003500. - Sture Sjöstedt, Dec 19 2017

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
Tanya Khovanova, Recursive Sequences
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
E. Keith Lloyd, The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles, Math. Gaz. vol 81 (1997), 231-243.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
William H. Richardson, Super-Heronian Triangles.
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A001075, A001353, A002194, A003500, A082841.

[3*Evaluate(ChebyshevSecond(n), 2): n in [0..40]]; // G. C. Greubel, Oct 10 2022

A005320:=3*z/(1-4*z+z**2); # Simon Plouffe in his 1992 dissertation
a:= n-> (Matrix([[3,0]]). Matrix([[4,1],[ -1,0]])^n)[1,2]: seq(a(n), n=0..50); # Alois P. Heinz, Aug 14 2008

LinearRecurrence[{4,-1},{0,3},40] (* Harvey P. Dale, Mar 04 2012 *)

Vec(3/(x^2-4*x+1)+O(x^99)) \\ Charles R Greathouse IV, Mar 05 2012

[3*chebyshev_U(n-1,2) for n in range(41)] # G. C. Greubel, Oct 10 2022

a(n) = (sqrt(3)/2)*( (2+sqrt(3))^n - (2-sqrt(3))^n ). - Antonio Alberto Olivares, Jan 17 2004
G.f.: 3*x/(1-4*x+x^2). - Harvey P. Dale, Mar 04 2012
a(n) = 3*A001353(n). - R. J. Mathar, Mar 14 2016

Typo in definition corrected by Johannes Boot, Feb 05 2009

A019824 Decimal expansion of sine of 15 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the imaginary part of i^(1/6). - Stanislav Sykora, Apr 25 2012

			0.258819045102520762348898837624048328349068901319930513814003207315...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mark B. Villarino, Legendre's Singular Modulus, arXiv:2002.07250 [math.HO], 2020.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 4

Cf. A002194, A020765, A101263.

RealDigits[ Sin[ Pi/12], 10, 111][[1]] (* Or *) RealDigits[(Sqrt[3] - 1)/(2 Sqrt[2]), 10, 111][[1]] (* Robert G. Wilson v *)
RealDigits[Sin[15 Degree],10,120][[1]] (* Harvey P. Dale, Jul 16 2016 *)

sin(Pi/12) \\ Charles R Greathouse IV, Apr 25 2012

Equals (sqrt(3)-1)/(2*sqrt(2)) = (A002194 -1) * A020765 = sin(Pi/12). - R. J. Mathar, Jun 18 2006
Equals 2F1(9/8,-1/8;1/2;3/4) / 2 = - 2F1(11/8,-3/8;1/2;3/4) / 2 = cos(5*Pi/12). - R. J. Mathar, Oct 27 2008
Equals sqrt(2 - sqrt(3))/2 = (1/2) * A101263. - Amiram Eldar, Aug 05 2020
Equals A019819 * A019894 + A019814 * A019889 = A019821 * A019896 + A019812 * A019887. - R. J. Mathar, Jan 27 2021
This^2 + A019884^2=1. - R. J. Mathar, Aug 31 2025
Smallest positive of the 4 real-valued roots of 16*x^4-16*x^2+1=0. - R. J. Mathar, Aug 31 2025
Equals 1/(4*A019884). - R. J. Mathar, Sep 05 2025

A072895 Least k for the Theodorus spiral to complete n revolutions.

Original entry on oeis.org

17, 54, 110, 186, 281, 396, 532, 686, 861, 1055, 1269, 1503, 1757, 2030, 2323, 2636, 2968, 3320, 3692, 4084, 4495, 4927, 5377, 5848, 6338, 6849, 7378, 7928, 8497, 9087, 9695, 10324, 10972, 11640, 12328, 13036, 13763, 14510, 15277, 16063, 16869

Offset: 1

Robert G. Wilson v, Jul 29 2002

nonn

"For n = 16 (which gives sqrt(17)) this sum is 351.15 degree, while for n = 17 the sum is 364.78 degree. That is, perhaps Theodorus stopped at sqrt(17) simply because for n > 16 his spiral started to overlap itself and the drawing became 'messy.'" Nahin p. 34.
There exists a constant c = 1.07889149832972311... such that b(n) = IntegerPart[(Pi*n + c)^2 - 1/6] differs at most by 1 from a(n) for all n>=1. At least for n<=4000 we indeed have a(n)=b(n). - Herbert Kociemba, Sep 12 2005
The preceding constant and function b(n) = a(n) for all n < 21001. - Robert G. Wilson v, Mar 07 2013; Update: b(n) = a(n) for all n < 10^9. - Herbert Kociemba, Jul 15 2013
The preceding constant, c, is actually is -K/2, where K is the Hlawka's Schneckenkonstante (A105459). - Robert G. Wilson v, Jul 10 2013

P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.
Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, page 35.
Paul J. Nahin, "An Imaginary Tale, The Story of [Sqrt(-1)]," Princeton University Press, Princeton, NJ. 1998, pgs 33-34.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, London, England, 1997, page 76.

Robert G. Wilson v, Table of n, a(n) for n = 1..21000
Edmund Hlawka, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math., 89 (1980) 19-44. [For a summary in English see the Davis reference, pp. 157-167.]
Herbert Kociemba, The Spiral of Theodorus

Cf. A002194, A105459.

s = 0; k = 1; lst = {}; Do[ While[s < (2Pi)n, (* change the value in the parentheses to change the angle *) s = N[s + ArcTan[1/Sqrt@k], 32]; k++]; AppendTo[lst, k - 1], {n, 50}]; lst (* Robert G. Wilson v, Oct 14 2012 *)
K = -2.15778299665944622; f[n_] := Floor[(n*Pi - K/2)^2 - 1/6]; Array[f, 41] (* Robert G. Wilson v, Jul 10 2013 *)
K = -2.1577829966594462209291427868295777235; a[n_] := Module[{a = -(K/2) + n Pi, b}, b = a^2 - 1/6; If[Floor[b] == Floor[b + 1/(144 a^2)], Floor[b], Undefined]] (* defined at least for all n < 10^9, Herbert Kociemba, Jul 15 2013 *)

a(n) = k means k is the least integer such that Sum_{i=1..k} arctan(1/sqrt(i)) > 2n*Pi.

a(n) = A137515(n) + 1. - Robert G. Wilson v, Feb 27 2013

A020784 Decimal expansion of 1/sqrt(27).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This is the minimum ripple factor for a third-order Chebyshev filter for which the generalized reflectionless topology needs no negative elements. - Matthew A. Morgan, Oct 18 2017

			0.1924500897298752548363829268339858185492005837567089586728674....

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.4.3 and 8.16, pp. 495, 527.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010482, A020760, A021031, A073010, A212886.

1/Sqrt(27); // G. C. Greubel, Feb 15 2018

RealDigits[1/Sqrt[27],10,200][[1]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)
realDigitsRecip[Sqrt[27]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Jul 05 2025 *)

1/sqrt(27) \\ G. C. Greubel, Feb 15 2018

Equals Sum_{k>=0} binomial(2*k,k) * k/16^k. - Amiram Eldar, Aug 02 2020
Equals sqrt(3)/9. - Stefano Spezia, Dec 24 2024
Equals 1/A010482 = A020760/3 = sqrt(A021031) = A073010/Pi = A212886/2. - Hugo Pfoertner, Dec 24 2024

A160390 Decimal expansion of sqrt(3) - 1.

Original entry on oeis.org

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

Offset: 0

Harry J. Smith, May 31 2009

Continued fraction expansion leads to the ternary digital root of n.

			0.732050807568877293527446341505872366942805253810380628055806979451933...

Harry J. Smith, Table of n, a(n) for n = 0..20000
Index entries for algebraic numbers, degree 2

Cf. A002194, A134451 (continued fraction).

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

RealDigits[Sqrt[3]-1,10,120][[1]] (* Harvey P. Dale, Dec 16 2016 *)

default(realprecision, 20080); x=10*(sqrt(3)-1); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b160390.txt", n, " ", d));

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

A019913 Decimal expansion of tangent of 15 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also, 2 - sqrt(3) = cotangent of 75 degrees. An equivalent definition of this sequence: decimal expansion of x < 1 satisfying x^2 - 4*x + 1 = 0. - Arkadiusz Wesolowski, Nov 29 2011
Multiplied by -1 (that is, -2 + sqrt(3)), this is one of three real solutions to x^3 = 15x + 4. The other two are 4 and -2 - sqrt(3), all of which can be found with Viete's formula. - Alonso del Arte, Dec 15 2012
Wentworth (1903) shows how to compute the tangent of 15 degrees to five decimal places by the laborious process of adding up the first few terms of Pi/12 + Pi^3/5184 + 2Pi^5/3732480 + 17Pi^7/11287019520 + ... - Alonso del Arte, Mar 13 2015
A quadratic integer. - Charles R Greathouse IV, Aug 27 2017
This is the radius of the largest sphere that can be placed in the space between a sphere of radius 1 and the corners of its circumscribing cube. - Amiram Eldar, Jul 11 2020

			0.2679491924311227064725536...

Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1). Princeton, New Jersey: Princeton University Press (1988): 22 - 23.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Willis F. Kern and James R. Bland, Solid Mensuration: With Proofs, 2nd ed., J. Wiley & Sons, Inc., New York, 1938. See pp. 91-92.
George Albert Wentworth, New Plane and Spherical Trigonometry, Surveying, and Navigation, Boston: The Atheneum Press (1903), p. 240.
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 2

Cf. A002194 (sqrt(3)).

RealDigits[N[Tan[15 Degree], 200]][[1]] (* Arkadiusz Wesolowski, Nov 29 2011 *)

2-sqrt(3) \\ Charles R Greathouse IV, Aug 27 2017

Equals Sum_{k>=1} binomial(2*k,k)/(6^k*(k+1)). - Amiram Eldar, Jul 11 2020
Equals exp(-arccosh(2)). - Amiram Eldar, Jul 06 2023
tan(Pi/12) = A019824 / A019884. - R. J. Mathar, Aug 31 2025

A020821 Decimal expansion of 1/8.

Original entry on oeis.org

1, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Also, decimal expansion of Integral_{x=0..oo} x^2/cosh(Pi*x) dx. - Bruno Berselli, Mar 20 2013
Also, decimal expansion of Sum_{i>=1} 1/9^i. - Bruno Berselli, Jan 03 2014
For any triangle ABC, sin(A/2) * sin(B/2) * sin(C/2) <= 1/8, equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 16 2022

			0.12500000000000000000...

Kiran S. Kedlaya, A < B, (1999), Problem 6.3, p. 6.
Srinivasa Ramanujan, Question 629, Journal of the Indian Mathematical Society, Vol. 7 (1918), p. 40.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A002194, A104956 (other trigonometric inequalities).

PadRight[{1, 2, 5}, 100] (* Paolo Xausa, Aug 27 2024 *)

Equals Sum_{k>=1} exp(-Pi*k^2) * (Pi*k^2 - 1/4) (Ramanujan, 1918). - Amiram Eldar, Jan 01 2025

A135611 Decimal expansion of sqrt(2) + sqrt(3).

Original entry on oeis.org

3, 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

Offset: 1

N. J. A. Sloane, Mar 03 2008

From Alexander R. Povolotsky, Mar 04 2008: (Start)
The value of sqrt(2) + sqrt(3) ~= 3.146264369941972342329135... is "close" to Pi. [See Borel 1926. - Charles R Greathouse IV, Apr 26 2014] We can get a better approximation by solving the equation: (2-x)^(1/(2+x)) + (3-x)^(1/(2+x)) = Pi.
Olivier Gérard finds that x is 0.00343476569746030039595770020414255107204742044644777... (End)
Another approximation to Pi is (203*sqrt(2)+ 197*sqrt(3))/200 = 3.1414968... - Alexander R. Povolotsky, Mar 22 2008
Shape of a sqrt(8)-extension rectangle; see A188640.  - Clark Kimberling, Apr 13 2011
This number is irrational, as instinct would indicate. Niven (1961) gives a proof of irrationality that requires first proving that sqrt(6) is irrational. - Alonso del Arte, Dec 07 2012
An algebraic integer of degree 4: largest root of x^4 - 10*x^2 + 1. - Charles R Greathouse IV, Sep 13 2013
Karl Popper considers whether this approximation to Pi might have been known to Plato, or even conjectured to be exact. - Charles R Greathouse IV, Apr 26 2014

			3.14626436994197234232913506571557044551247712918732870...

Emile Borel, Space and Time (1926).
Ivan Niven, Numbers: Rational and Irrational. New York: Random House for Yale University (1961): 44.
Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): 44.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Susan Landau, Simplification of nested radicals, SIAM Journal on Computing 21.1 (1992): 85-110. See page 85. [Do not confuse this paper with the short FOCS conference paper with the same title, which is only a few pages long.]
Burkard Polster, Irrational roots, Mathologer video (2018)
Karl Popper, The Open Society and Its Enemies, 1962.
Index entries for algebraic numbers, degree 4

Cf. A188640, A089078, A002193, A002194, A010464, A340616.

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

evalf(add(sqrt(ithprime(i)), i=1..2), 118);  # Alois P. Heinz, Jun 13 2022

r = 8^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A135611 *)
ContinuedFraction[t, 120]  (* A089078 *)
RealDigits[Sqrt[2] + Sqrt[3], 10, 100][[1]] (* G. C. Greubel, Oct 22 2016 *)

sqrt(2)+sqrt(3) \\ Charles R Greathouse IV, Sep 13 2013

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

Sqrt(2)+sqrt(3) = sqrt(5+2*sqrt(6)). [Landau, p. 85] - N. J. A. Sloane, Aug 27 2018
Equals 1/A340616. - Hugo Pfoertner, May 08 2024
Equals Product_{k>=0} (((4*k + 1)*(12*k + 11))/((4*k + 3)*(12*k + 1)))^(-1)^k. - Antonio Graciá Llorente, May 22 2024

A152623 Decimal expansion of 3/2.

Original entry on oeis.org

1, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Oct 30 2009

Sum of the inverses of the tetrahedral numbers (A000292). - Michael B. Porter, Nov 27 2017

For any triangle ABC, cos A + cos B + cos C <= 3/2; equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 17 2022

			1.5000000000000000000000000000000000000000000000000000000000...

Kiran S. Kedlaya, A < B, (1999), Problem 6.2, p. 6.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000292 (tetrahedral numbers).
Sums of inverses: A002117 (cubes), A175577 (octahedral numbers), A295421 (dodecahedral numbers), A175578 (icosahedral numbers).
Cf. A002194, A020821, A104956 (other trigonometric inequalities).

3/2. \\ Charles R Greathouse IV, Jan 10 2022

cp's OEIS Frontend

A131595 Decimal expansion of 3*(sqrt(25 + 10*sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A005320 a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A019824 Decimal expansion of sine of 15 degrees.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A072895 Least k for the Theodorus spiral to complete n revolutions.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A020784 Decimal expansion of 1/sqrt(27).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A160390 Decimal expansion of sqrt(3) - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

A131595 Decimal expansion of 3(sqrt(25 + 10sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.