A073000 Decimal expansion of arctangent of 1/2.

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

Offset: 0

Author

Robert G. Wilson v, Aug 03 2002

Keywords

• cons
• nonn

Comments

The angle at which you must shoot a cue ball on a standard pool table so that it will strike all four sides and return to its origin. [Barrow] - Robert G. Wilson v, Nov 29 2015

Examples

			Arctan(1/2)
=0.463647609000806116214256231461214402028537054286120263810933088720197864165... radians
=26°.56505117707798935157219372045329467120421429964522102798601631528806582148474...
=26°33'.9030706246793610943316232271976802722528579787132616791609789172839492890...
=26°33'54".184237480761665659897393631860816335171478722795700749658735037036957...
complement = 63°.43494882292201064842780627954670532879578570035477897201398368471...
supplement = 153°.4349488229220106484278062795467053287957857003547789720139836847...
		

References

• John D. Barrow, One Hundred Essential Things You Didn't Know You Didn't Know, W. W. Norton & Co., NY & London, 2008.
• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 242.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Peter Bala, New series for old functions.
• R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv preprint arXiv:1309.3705 [math.MG], 2013.
• Simon Plouffe, arctan(1/2).
• Index entries for transcendental numbers

Crossrefs

Cf. A001622, A002162, A002391, A105531, A254619.

Programs

Maple

evalf(arctan(0.5)) ; # R. J. Mathar, Aug 22 2013

Mathematica

RealDigits[ ArcTan[1/2], 10, 110] [[1]]

PARI

default(realprecision,2000); atan(1/2) \\ Anders Hellström, Nov 30 2015

Formula

Equals Pi/2 - A105199 = A019669-A105199. - R. J. Mathar, Aug 21 2013

From Peter Bala, Feb 04 2015: (Start)

Arctan(1/2) = 1/2*Sum_{k >= 0} (-1)^k/((2*k + 1)*4^k).

Define a pair of integer sequences A(n) = 4^n*(2*n + 1)!/n! and B(n) = A(n)*Sum_{k = 0..n} (-1)^k/((2*k + 1)*4^k). Both sequences satisfy the same second order recurrence equation u(n) = (12*n + 10)*u(n-1) + 16*(2*n - 1)^2*u(n-2). From this observation we obtain the continued fraction expansion 2*arctan(1/2) = 1 - 2/(24 + 16*3^2/(34 + 16*5^2/(46 + ... + 16*(2*n - 1)^2/((12*n + 10) + ...)))). See A002391, A105531 and A002162 for similar expansions.

Arctan(1/2) = 2/5 * Sum_{k >= 0} (4/5)^k/((2*k + 1)*binomial(2*k,k)).

Define a pair of integer sequences C(n) = 5^n*(2*n + 1)!/n! and D(n) = C(n)*Sum_{k = 0..n} (4/5)^k/((2*k + 1)*binomial(2*k,k)). Both sequences satisfy the same second order recurrence equation u(n) = (24*n + 10)*u(n-1) - 40*n*(2*n - 1)^2*u(n-2). From this observation we obtain the continued fraction expansion 5/2*arctan(1/2) = 1 + 4/(30 - 240/(58 - 600/(82 - ... - 40*n*(2*n - 1)/((24*n + 10) - ... )))).

Arctan(1/2) = 2/25 * Sum_{k >= 0} (24*k + 17)*(4/5)^(2*k)/( (4*k + 1)*(4*k + 3)*binomial(4*k,2*k) ).

Arctan(1/2) = 2/125 * Sum_{k >= 0} (1116*k^2 + 1446*k + 433)*(4/5)^(3*k)/( (6*k + 1)*(6*k + 3)*(6*k + 5)*binomial(6*k,3*k) ). (End)

Equals Integral_{x = 0..oo} exp(-2*x)*sin(x)/x dx. - Peter Bala, Nov 05 2019

Equals 2 * arccot(phi^3), where phi is the golden ratio (A001622). - Amiram Eldar, Jul 06 2023

Equals Sum_{n >= 1} i/(n*P(n, 2*i)*P(n-1, 2*i)) = (1/2)*Sum_{n >= 1} (-1)^(n+1)*4^n/(n*A098443(n)*A098443(n-1)), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The n-th summand of the series is O( 1/(3 + 2*sqrt(2))^n ). - Peter Bala, Mar 16 2024