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A019670 Decimal expansion of Pi/3. 23
1, 0, 4, 7, 1, 9, 7, 5, 5, 1, 1, 9, 6, 5, 9, 7, 7, 4, 6, 1, 5, 4, 2, 1, 4, 4, 6, 1, 0, 9, 3, 1, 6, 7, 6, 2, 8, 0, 6, 5, 7, 2, 3, 1, 3, 3, 1, 2, 5, 0, 3, 5, 2, 7, 3, 6, 5, 8, 3, 1, 4, 8, 6, 4, 1, 0, 2, 6, 0, 5, 4, 6, 8, 7, 6, 2, 0, 6, 9, 6, 6, 6, 2, 0, 9, 3, 4, 4, 9, 4, 1, 7, 8, 0, 7, 0, 5, 6, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

With an offset of zero, also the decimal expansion of Pi/30 ~ 0.104719... which is the average arithmetic area <S_0> of the 0-winding sectors enclosed by a closed Brownian planar path, of a given length t, according to Desbois, p.1. - Jonathan Vos Post, Jan 23 2011

Polar angle (or apex angle) of the cone that subtends exactly one quarter of the full solid angle. See comments in A238238. - Stanislav Sykora, Jun 07 2014

a = Product_{n >= 1} A016910(n) / A136017(n). -  Fred Daniel Kline, Jun 09 2016

60 degrees in radians. - M. F. Hasler, Jul 08 2016

LINKS

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Kunle Adegoke, Infinite arctangent sums involving Fibonacci and Lucas numbers, arXiv:1603.08097 [math.NT], 2016.

J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.

Jean Desbois, Stephane Ouvry, Algebraic and arithmetic area for m planar Brownian paths, arXiv:1101.4135 [math-ph], Jan 21, 2011.

FORMULA

A third of A000796, a sixth of A019692, the square root of A100044.

Sum_{k >= 0} (-1)^k/(6k+1) + (-1)^k/(6k+5). - Charles R Greathouse IV, Sep 08 2011

Product_{k >= 1}(1-(6k)^{-2})^{-1}. - Fred Daniel Kline, May 30 2013

From Peter Bala, Feb 05 2015: (Start)

Pi/3 = Sum {k >= 0} binomial(2*k,k)*1/(2*k + 1)*(1/16)^k = 2F1(1/2,1/2;3/2;1/4). Similar series expansions hold for Pi^2 (A002388), Pi^3 (A091925) and Pi/(2*sqrt(2)) (A093954.)

The integer sequences A(n) := 4^n*(2*n + 1)! and B(n) := A(n)*( Sum {k = 0..n} binomial(2*k,k)*1/(2*k + 1)*(1/16)^k ) both satisfy the second order recurrence equation u(n) = (20*n^2 + 4*n + 1)*u(n-1) - 8*(n - 1)*(2*n - 1)^3*u(n-2). From this observation we can obtain the continued fraction expansion Pi/3 = 1 + 1/(24 - 8*3^3/(89 - 8*2*5^3/(193 - 8*3*7^3/(337 - ... - 8*(n - 1)*(2*n - 1)^3/((20*n^2 + 4*n + 1) - ... ))))). Cf. A002388 and A093954. (End)

Equals Sum_{k >= 1} arctan(sqrt(3)*L(2k)/L(4k)) where L=A000032. See also A005248 and A056854. - Michel Marcus, Mar 29 2016

Equals Integral_{x=-inf..inf} sech(x)/3 dx. - Ilya Gutkovskiy, Jun 09 2016

From Peter Bala, Nov 16 2016: (Start)

Euler's series transformation applied to the series representation Pi/3 = Sum_{k >= 0} (-1)^k/(6*k + 1) + (-1)^k/(6*k + 5) given above by Greathouse produces the faster converging series Pi/3 = 1/2 * Sum_{n >= 0} 3^n*n!*( 1/(Product_{k = 0..n} (6*k + 1)) + 1/(Product_{k = 0..n} (6*k + 5)) ).

The series given above by Greathouse is the case n = 0 of the more general result Pi/3 = 9^n*(2*n)! * Sum_{k >= 0} (-1)^(k+n)*( 1/(Product_{j = -n..n} (6*k + 1 + 6*j)) + 1/(Product_{j = -n..n} (6*k + 5 + 6*j)) ) for n = 0,1,2,.... Cf. A003881. See the example section for notes on the case n = 1.(End)

Pi/3 = Product_{p>=5, p prime} p/sqrt(p^2-1). - Dimitris Valianatos, May 13 2017

EXAMPLE

Pi/3 = 1.04719755119659774615421446109316762806572313312503527365831486...

From Peter Bala, Nov 16 2016: (Start)

Case n = 1. Pi/3 = 18 * Sum_{k >= 0} (-1)^(k+1)( 1/((6*k - 5)*(6*k + 1)*(6*k + 7)) + 1/((6*k - 1)*(6*k + 5)*(6*k + 11)) ).

Using the methods of Borwein et al. we can find the following asymptotic expansion for the tails of this series: for N divisible by 6 there holds Sum_{k >= N/6} (-1)^(k+1)( 1/((6*k - 5)*(6*k + 1)*(6*k + 7)) + 1/((6*k - 1)*(6*k + 5)*(6*k + 11)) ) ~ 1/N^3 + 6/N^5 + 1671/N ^7 - 241604/N^9 + ..., where the sequence [1, 0, 6, 0, 1671, 0, -241604, 0,...] is the sequence of coefficients in the expansion of 1/18*cosh(2*x)/cosh(3*x) * sinh(3*x)^2 = x^2/2! + 6*x^4/4! + 1671*x^6/6! - 241604*x^8/8! + .... Cf. A024235, A278080 and A278195. (End)

MATHEMATICA

RealDigits[N[Pi/3, 6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

PROG

(PARI) Pi/3 \\ Charles R Greathouse IV, Sep 08 2011

(PARI) N=150; default(realprecision, 3+N); digits(Pi*10^N\3) \\ M. F. Hasler, Jul 08 2016

CROSSREFS

Cf. A000796 (Pi), A019669 (Pi/2), A019692 (2*Pi), A122952 (3*Pi), A003881(Pi/4), A137914, A238238, A002388, A091925, A093954, A016910, A019694, A136017.

Sequence in context: A021216 A085508 A198347 * A093436 A082169 A209634

Adjacent sequences:  A019667 A019668 A019669 * A019671 A019672 A019673

KEYWORD

nonn,cons,changed

AUTHOR

N. J. A. Sloane, Dec 11 1996

STATUS

approved

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Last modified May 24 18:18 EDT 2017. Contains 286997 sequences.