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a(n) ~ (2+2^(n - 3/4) * (1 + sqrt(2))^(n + 1/sqrt((4-2*) / sqrt(2))*Pi*n). - Vaclav Kotesovec, Oct 05 2012, simplified Jan 31 2023

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This sequence gives the integer part of an integral approximation to Pi, and also appears in Frits Beukers's "A Rational Approach to Pi" (cf. Links, Example). Despite quality M ~ 0.9058... reported by Beukers, measurements between n = 10000 and 30000 lead to a contentious quality estimate, M ~ 0.79..., at the 99% confidence level. In "Searching for Apéry-Style Miracles" Doron Zeilberger Quotes that M = 0.79119792... and also gives a closed form. The same rational approximation to Pi also follows from time integration on a quartic Hamiltonian surface, 2*H=(q^2+p^2)*(1-4*q*(q-p)). - Bradley Klee, Jul 19 2018, updated Mar 17 21092019

seq( sumadd('binomial(2*k, k)*binomial(k, n-k)', ', k'=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001

A006139 := n -> 2^n*hypergeom([-n/2, 1/2-n/2], [1], 2); :

seq(round(evalfsimplify(A006139(n), 99)), n=0..29); # Peter Luschny, Sep 18 2014

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F. Frits Beukers, <a href="https://dspace.library.uu.nl/handle/1874/26398">A rational approach to Pi</a>, Nieuw archief voor wiskunde 5/1 No. 4, December 2000, p. 377.

D. Dario Castellanos, <a href="http://www.fq.math.ca/Scanned/27-5/castellanos.pdf">A generalization of Binet's formula and some of its consequences</a>, Fib. Quart., 27 (1989), 424-438.

M. Maciej Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv:1410.5747 [math.CO], 2014.

S. Shalosh B. Ekhad and D. Doron Zeilberger, <a href="https://arxiv.org/abs/1405.4445">Searching for Apéry-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm]</a>, arXiv:1405.4445 [math.NT], 2014.

B. Bradley Klee, <a href="http://demonstrations.wolfram.com/ApproximatingPiWithTrigonometricPolynomialIntegrals/">Approximating Pi with Trigonometric-Polynomial Integrals</a>, Wolfram Demonstrations, July 27, 2018.

Cf. A001850, A002426, A036442, A084600-A084606, A084608-A084615.

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S. B. Ekhad and D. Zeilberger, <a href="https://arxiv.org/abs/1405.4445">Searching for Apéry-Style Miracles</a>, arXiv:1405.4445 [math.NT], 2014.

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