A280813 Denominators 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)).

7, 15015, 137287920, 235953517800, 8548690331301120, 67462193289708771840, 161102819285860855603200, 6305423381881718760060595200, 7411866941185812791748757094400, 28422996899365886608045972478361600, 24827411794278189209115835981312819200

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, A280812 (numerators).

Sequence in context: A333338 A131676 A344532 * A203685 A134645 A327840

Adjacent sequences: A280810 A280811 A280812 * A280814 A280815 A280816

Keyword

nonn,frac

Author

Seiichi Manyama, Jan 08 2017

Status

approved