A072173 a(n) = (2*n+1)*239^(2*n+1).

239, 40955757, 3899056325995, 311805194956024553, 22899374409963958061031, 1598709646931895970271741029, 107923510786468980575690686466147, 7113114068808339968612339655730133985, 460482613887654678993386180604955781138143, 29397724727626925615108413436728112018437968221

Offset: 0

Author

N. J. A. Sloane, Jun 30 2002

Keywords

• nonn
• easy

Comments

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

References

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

Links

• Colin Barker, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (114242,-3262808641).

Crossrefs

Cf. A072172.

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

Programs

GAP

List([0..10], n-> (2*n+1)*239^(2*n+1)); # G. C. Greubel, Aug 26 2019

Magma

[(2*n+1)*239^(2*n+1): n in [0..10]]; // G. C. Greubel, Aug 26 2019

Maple

seq((2*n+1)*239^(2*n+1), n = 0..10); # G. C. Greubel, Aug 26 2019

Mathematica

Table[(2*n+1)*239^(2*n+1), {n,0,10}] (* G. C. Greubel, Aug 26 2019 *)

PARI

Vec(239*(1+57121*x)/(1-57121*x)^2 + O(x^10)) \\ Colin Barker, Aug 25 2016

PARI

vector(10, n, (2*n-1)*239^(2*n-1)) \\ G. C. Greubel, Aug 26 2019

Sage

[(2*n+1)*239^(2*n+1) for n in (0..10)] # G. C. Greubel, Aug 26 2019

Formula

From Colin Barker, Aug 25 2016: (Start)

a(n) = 114242*a(n-1) - 3262808641*a(n-2) for n>1.

G.f.: 239*(1+57121*x) / (1-57121*x)^2. (End)

E.g.f.: m*(1+2*m^2*x)*exp(m^2*x), where m=239. - G. C. Greubel, Aug 26 2019