A105790 Number of bisections to an inscribed triangle to approximate Pi (A000796) to n decimal digits of accuracy.

1, 4, 4, 6, 8, 9, 11, 13, 14, 16, 17, 19, 21, 23, 25, 26, 27, 30, 31, 33, 34, 36, 38, 40, 41, 43, 45, 46, 47, 49, 53, 53, 54, 56, 58, 60, 61, 62, 65, 66, 67, 70, 71, 72, 75, 76, 78, 80, 83, 83, 84, 87, 89, 89, 91, 93, 94, 96, 98, 99, 100, 103, 105, 107, 107, 109, 112, 112

Offset

1,2

References

Howard Anton, Irl C. Bivens and Stephen L. Davis, Calculus, Early Transcendentals, 7th Edition, John Wiley & Sons, Inc., NY, Section 6.1 An Overview of the Area Problem, page 372-377.

William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton, NJ 2005, page 56-57.

Links

Table of n, a(n) for n=1..68.

Formula

a(n) = 3*2^n*sqrt(2- sqrt(2+ sqrt(2+ ... sqrt(2+ sqrt(3))...))).

A(n) in Table 6.1.1 = Sin( 2Pi/n )*n/2. - Anton.

Mathematica

$MaxExtraPrecision =128; p=RealDigits[ Pi, 10, 100][[1]]; f[n_] := 3*2^(n)*Sqrt[2 - Nest[ Sqrt[2 + # ] &, Sqrt[3], n - 1]]; g[n_] := Block[{k = 1, q = Take[p, n + 1]}, While[ Take[ RealDigits[ f[k], 10, 100][[1]], n + 1] != q, k++ ]; k]; Table[ g[n], {n, 69}]

Crossrefs

Cf. A000796.

Sequence in context: A294021 A200267 A273308 * A338895 A090682 A011982

Adjacent sequences: A105787 A105788 A105789 * A105791 A105792 A105793

Keyword

nonn,base

Author

Robert G. Wilson v, Apr 20 2005

Status

approved