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A248180 Least k such that r - sum{1/C(2h+1,h), h = 0..k} < 1/2^n, where r = (2/27)*(9 + 2*sqrt(3)*Pi). 4
1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This sequence gives a measure of the convergence rate of sum{1/C(2h+1,h), h = 0..k}.  Since a(n+1) - a(n) is in {0,1} for n >= 0, the sequences A248195 and A248196 partition the positive integers.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..3000

EXAMPLE

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

n ... r - s(n) ..... 1/2^n

0 ... 0.47289 ...... 1

1 ... 0.139466 ..... 0.5

2 ... 0.0394664 .... 0.25

3 ... 0.010895 ..... 0.125

4 ... 0.00295845 ... 0.0625

a(3) = 2 because r - s(2) < 1/8 < r - s(1).

MATHEMATICA

$MaxExtraPrecision = Infinity;

z = 300; p[k_] := p[k] = Sum[1/Binomial[2 h + 1, h], {h, 0, k}] ;

r = Sum[1/Binomial[2 h + 1, h], {h, 0, Infinity}]  (* A248179 *)

r = 2/27 (9 + 2 Sqrt[3] \[Pi]); N[r, 20]

N[Table[r - p[n], {n, 0, z/10}]]

f[n_] := f[n] = Select[Range[z], r - p[#] < 1/2^n &, 1]

u = Flatten[Table[f[n], {n, 0, z}]]  (* A248180 *)

Flatten[Position[Differences[u], 0]] (* A248195 *)

Flatten[Position[Differences[u], 1]] (* A248196 *)

CROSSREFS

Cf. A248179, A248195, A248196.

Sequence in context: A109728 A157271 A025162 * A025161 A021895 A025160

Adjacent sequences:  A248177 A248178 A248179 * A248181 A248182 A248183

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Oct 03 2014

STATUS

approved

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Last modified November 22 10:59 EST 2019. Contains 329389 sequences. (Running on oeis4.)