A263498 -id:A263498

Search: a263498 -id:a263498

Displaying 1-2 of 2 results found.

page 1

A275486

Decimal expansion of Pi_3, the analog of Pi for generalized trigonometric functions of order p=3.

+10
6

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The area of the circumcircle of a unit-area equilateral triangle. - Amiram Eldar, Aug 13 2020

LINKS

Table of n, a(n) for n=1..102.

David Edmunds and Jan Lang, Generalizing trigonometric functions from different points of view, 2009.

Shingo Takeuchi, A new form of the generalized complete elliptic integrals, arXiv:1411.4778 [math.CA], 2014.

FORMULA

Pi_3 = 2*Pi/(3*sin(Pi/3)) = 2/3 * gamma(1/3) * gamma(2/3) = 4*Pi/(3 * sqrt(3)).

Pi_3 = Sum_{n>=1} 4/(9*n^2 - 9*n + 2).

Pi_3 = 2*Integral_{0..1} (1-x^3)^(-1/3) dx.

Equals 1 + A263498.

Equals Integral_{x=0..oo} 1/(1 + x^(3/2)) dx. - Amiram Eldar, Aug 13 2020

Equals Product_{p prime} (1 + Kronecker(-3, p)/p)^(-1) = Product_{p prime != 3} (1 - (-1)^(p mod 3)/p)^(-1). - Amiram Eldar, Nov 06 2023

EXAMPLE

2.41839915231229046745877101018954097637875499745698743409317991385...

MATHEMATICA

RealDigits[4 Pi/(3 Sqrt[3]), 10, 102][[1]]

PROG

(PARI) 4*Pi/sqrt(27) \\ Charles R Greathouse IV, Aug 01 2016

CROSSREFS

Cf. A240935 (reciprocal), A263498.

KEYWORD

nonn,cons,easy

AUTHOR

Jean-François Alcover, Jul 30 2016

STATUS

approved

A271563

Decimal expansion of Sum_{j>=0} Sum_{i>=0} (-1/4)^i*(-1)^j*binomial(2i,i)/((2j+1)(i+2j+2)).

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..84.

FORMULA

Equals (Pi - 2*sqrt(1+sqrt(2)) * arctan(2*sqrt(2+10*sqrt(2))/7)) / sqrt(2). - Vaclav Kotesovec, Apr 10 2016

EXAMPLE

0.3445436367923706403320533879002043065894259746135921255085777...

MAPLE

evalf((Pi - 2*sqrt(1+sqrt(2)) * arctan(2*sqrt(2+10*sqrt(2))/7)) / sqrt(2), 120); # Vaclav Kotesovec, Apr 10 2016

MATHEMATICA

RealDigits[(Pi - 2*Sqrt[1 + Sqrt[2]] * ArcTan[(2/7)*Sqrt[2 + 10*Sqrt[2]]])/Sqrt[2], 10, 120][[1]]

N[Sum[Sum[((-1)^(i + j) 4^-i Binomial[2 i, i])/((1 + 2 j) (2 + i + 2 j)), {i, 0, Infinity}], {j, 0, Infinity}]]

PROG

(PARI) (Pi - 2*sqrt(1+sqrt(2)) * atan(2*sqrt(2+10*sqrt(2))/7)) / sqrt(2)

CROSSREFS

Decimal expansions of hypergeometric series: A244844, A263353, A263354, A263490, A263491, A263492, A263493, A263494, A263495, A263496, A263497, A263498, A229837, A268813, A086731, A002117, A076788.

KEYWORD

cons,nonn

AUTHOR

John M. Campbell, Apr 09 2016

STATUS

approved

page 1

Search completed in 0.007 seconds