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

Author

Robert G. Wilson v, Apr 20 2005

Keywords

• nonn
• base

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.

Crossrefs

Cf. A000796.

Programs

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}]

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.