A105532 -id:A105532 - cp's OEIS frontend

A072172 a(n) = (2n+1)5^(2*n+1).

5, 375, 15625, 546875, 17578125, 537109375, 15869140625, 457763671875, 12969970703125, 362396240234375, 10013580322265625, 274181365966796875, 7450580596923828125, 201165676116943359375, 5401670932769775390625, 144354999065399169921875

Offset: 0

N. J. A. Sloane, Jun 30 2002

J. Machin (died 1751) used Pi/4 = 4*Sum_{n=0..inf} (-1)^n/((2*n+1)*5^(2*n+1)) - Sum_{n=0..inf} (-1)^n/((2*n+1)*239^(2*n+1)) to calculate Pi to 100 decimal places.

H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 73

Colin Barker, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (50,-625).

Cf. A072173.

Cf. A157332. - Jaume Oliver Lafont, Mar 03 2009

List([0..20], n-> (2*n+1)*5^(2*n+1)); # G. C. Greubel, Aug 26 2019

[(2*n+1)*5^(2*n+1): n in [0..20]]; // G. C. Greubel, Aug 26 2019

seq((2*n+1)*5^(2*n+1), n=0..20); # G. C. Greubel, Aug 26 2019

Table[(2*n+1)*5^(2*n+1), {n,0,20}] (* G. C. Greubel, Aug 26 2019 *)

Vec(5*(1+25*x)/(1-25*x)^2 + O(x^20)) \\ Colin Barker, Aug 25 2016

vector(20, n, (2*n-1)*5^(2*n-1) ) \\ G. C. Greubel, Aug 26 2019

[(2*n+1)*5^(2*n+1) for n in (0..20)] # G. C. Greubel, Aug 26 2019

From Colin Barker, Aug 25 2016: (Start)
a(n) = 50*a(n-1) - 625*a(n-2) for n>1.
G.f.: 5*(1+25*x)/(1-25*x)^2.
(End)
From Ilya Gutkovskiy, Aug 25 2016: (Start)
E.g.f.: 5*(1 + 50*x)*exp(25*x).
Sum_{n>=0} 1/a(n) = arctanh(1/5) = 0.2027325540540821...
Sum_{n>=0} (-1)^n/a(n) = arctan(1/5) = A105532 (End)

A195769 Decimal expansion of arctan(5).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 24 2011

			arctan(5) = 1.373400766945015860861271926444961148650999...

Index entries for transcendental numbers

Cf. A105532, A195771, A195772.

r = 5;
N[ArcTan[r], 100]
RealDigits[%] (* A195769 *)
N[ArcCot[r], 100]
RealDigits[%] (* A105532 *)
N[ArcSec[r], 100]
RealDigits[%] (* A195771 *)
N[ArcCsc[r], 100]
RealDigits[%] (* A195772 *)

atan(5) \\ Charles R Greathouse IV, Nov 20 2024

Equals arcsin(5/sqrt(26)) = arccos(1/sqrt(26)). - Amiram Eldar, Jul 11 2023

A105534 Decimal expansion of arctan 1/239.

Original entry on oeis.org

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

Offset: 0

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005

Comment from Frank Ellermann, Mar 01 2020: (Start)
8*A195790 - arctan( 1/239 ) - 4*arctan( 1/515 ) = Pi/4 (Meissel, Klingenstierna).
12*arctan( 1/18 ) + 8*arctan( 1/57 ) - 5*arctan( 1/239 ) = Pi/4 (Gauss). (End)

			0.0041840760020747238645382149...

D. H. Lehmer, On Arccotangent Relations for π, The American Mathematical Monthly, Vol. 45, No. 10 (Dec., 1938), pp. 657-664.
Eric Weisstein's World of Mathematics, Machin-Like Formulas
Index entries for transcendental numbers

Cf. A003881 (Pi/4), A021243 (1/239), A105532 (arctan 1/5), A195790 (arccot 10).

len = 103; n = RealDigits[N[ArcTan[1/239], len]]; PadLeft[First@ n, len + Abs@ Last@ n] (* Michael De Vlieger, Sep 14 2015 *)
Join[{0,0},RealDigits[ArcTan[1/239],10,120][[1]]] (* Harvey P. Dale, Apr 29 2016 *)

atan(1/239) \\ Michel Marcus, Sep 24 2014

4*A105532 - arctan(1/239) = Pi/4 (Machin's formula).

arctan(1/239) = Sum_{n >= 1} i/(n*P(n, 239*i)*P(n-1, 239*i)) = 1/239 - 1/40955996 + 1/8773020079176 - 1/1948832181801673304 + 4/1753293766205137615850855 - ..., where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. - Peter Bala, Mar 21 2024

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A072172 a(n) = (2*n+1)*5^(2*n+1).

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A195769 Decimal expansion of arctan(5).

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A105534 Decimal expansion of arctan 1/239.

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Formula

A072172 a(n) = (2n+1)5^(2*n+1).