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

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

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..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.

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..700
• Index entries for linear recurrences with constant coefficients, signature (50,-625).

Crossrefs

Cf. A072173.

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

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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

PARI

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

PARI

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

Sage

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

Formula

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)