A215746 Numerator of Sum_{i=0..n} (-1)^i*4/(2*i + 1).

4, 8, 52, 304, 1052, 10312, 147916, 135904, 2490548, 44257352, 47028692, 1023461776, 5385020324, 15411418072, 467009482388, 13895021563328, 14442004718228, 13926277743608, 533322720625196, 516197940314096, 21831981985010836, 911392701638017048, 937558224301357108

Offset: 0

Author

Alonso del Arte, Aug 22 2012

Keywords

• nonn
• easy
• frac

Comments

Denominator of the sum divides A025547(n+1), but is not always equal to it: the first exception is n = 32.

x(n) = Sum_{i=0..n} (-1)^i*4/(2*i+1) very slowly converges to Pi, with x(n) > Pi when n is even and x(n) < Pi when n is odd.

Examples

			a(2) = 52 because 4 - 4/3 + 4/5 = 60/15 - 20/15 + 12/15 = 52/15.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..229
• Eric W. Weisstein, MathWorld: Pi formulas

Crossrefs

Cf. A007509.

Programs

Maple

N:= 100; # to get terms up to a[N]
T[0]:= 4;
A215746[0]:= 4;
for i from 1 to N do
  T[i]:= T[i-1] + (-1)^i*4/(2*i+1);
  A215746[i]:= numer(T[i])
od:
[seq](A215746[i],i=0..N); # Robert Israel, Apr 27 2014

Mathematica

Table[Numerator[Sum[(-1)^i 4/(2i + 1), {i, 0, n}]], {n, 0, 39}]

Extensions

Definition and comments corrected by Robert Israel, Apr 27 2014