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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)). 2

%I #21 Apr 25 2022 08:07:01

%S 7,15015,137287920,235953517800,8548690331301120,67462193289708771840,

%T 161102819285860855603200,6305423381881718760060595200,

%U 7411866941185812791748757094400,28422996899365886608045972478361600,24827411794278189209115835981312819200

%N 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)).

%C 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.

%H Seiichi Manyama, <a href="/A280813/b280813.txt">Table of n, a(n) for n = 1..249</a>

%H Jean-Christophe Pain, <a href="https://arxiv.org/abs/2204.10693">Successive approximations of Pi using Euler Beta functions</a>, arXiv:2204.10693 [math.HO], 2022. See Table 1 p. 3.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80">Proof that 22/7 exceeds Pi</a>

%e 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.

%e 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.

%Y Cf. A000796, A280812 (numerators).

%K nonn,frac

%O 1,1

%A _Seiichi Manyama_, Jan 08 2017

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