A280812 Numerators of 4 * Sum_{k=0..3*n-1} (-1)^k/(2*k+1) + (-1)^(n+1) * Sum_{k=0..2*n-1} (-1)^k/(2^(2*n-k-2) * (8*n-k-1) * binomial(8*n-k-2, 4*n+k)).

22, 47171, 431302721, 741269838109, 26856502742629699, 211938730834003723543, 506119433541064524255449, 19809071774292917047896724979, 23285066731814401580687501596643, 89293478252053341114758995682016773

Offset

1,1

Comments

1/(2^(2*n-1) * (8*n+1) * binomial(8*n, 4*n)) < 1/2^(2*n-2) * Integral_{x=0..1} (x^(4*n) * (1-x)^(4*n))/(1+x^2) dx < 1/(2^(2*n-2) * (8*n+1) * binomial(8*n, 4*n)). So b(n) = 4 * Sum_{k=0..3*n-1} (-1)^k/(2*k+1) + (-1)^(n+1) * Sum_{k=0..2*n-1} (-1)^k/(2^(2*n-k-2) * (8*n-k-1) * binomial(8*n-k-2, 4*n+k)) is nearly Pi. And the limit of b(n) is Pi.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..249

Jean-Christophe Pain, Successive approximations of Pi using Euler Beta functions, arXiv:2204.10693 [math.HO], 2022. See Table 1 p. 3.

Wikipedia, Proof that 22/7 exceeds Pi

Example

1/1260 < 1/2^0 * Integral_{x=0..1} (x^4 * (1-x)^4)/(1+x^2) dx < 1/630. So 1/1260 < 22/7 - Pi < 1/630.
1/1750320 < 1/2^2 * Integral_{x=0..1} (x^8 * (1-x)^8)/(1+x^2) dx < 1/875160. So 1/1750320 < Pi - 47171/15015 < 1/875160.

Crossrefs

Cf. A000796, A280813 (denominators).

Sequence in context: A191946 A221639 A078398 * A060619 A362902 A238635

Adjacent sequences: A280809 A280810 A280811 * A280813 A280814 A280815

Keyword

nonn,frac

Author

Seiichi Manyama, Jan 08 2017

Status

approved