A248148 - OEIS

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A248148

Least k such that r - sum{1/Binomial[2h, h], h = 0..k} < 1/3^n, where r = 1/3 + 2*Pi/Sqrt(243).

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 19, 20, 21, 22, 23, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55

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OFFSET

1,2

COMMENTS

It is well known that sum{1/Binomial[2h, h], h = 0..infinity} = r (approximately 0.7363998); this sequence gives a measure of the convergence rate. It appears that a(n+1) - a(n) is in {0,1} for n >= 1.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..2000

EXAMPLE

Let s(n) = sum{1/Binomial[2h, h], h = 0..n}. Approximations are shown here:

n ... r - s(n) ..... 1/3^n

1 ... 0.2364 ....... 0.33333

2 ... 0.0697332 .... 0.11111

3 ... 0.0197332 .... 0.037037

4 ... 0.00544748 ... 0.012345

5 ... 0.00147922 ... 0.004115

a(3) = 3 because r - s(3) < 1/27 < r - s(2).

MATHEMATICA