A221706 Kochanski approximates to sqrt(2) starting with R_0=3, S_0=2.

2, 4, 4, 15, 17, 77, 101, 119, 143, 250, 362, 1401, 31168, 88629, 184654, 259251, 298769, 4196069, 38538873, 616984562, 1975413034, 5345718056, 27843871196, 54516286512, 334398528973, 445879679625, 495957494385, 2450869042060, 2629541150528, 4088114099884

Offset: 0

Author

N. J. A. Sloane, Jan 23 2013

Keywords

• nonn

Links

• Henryk Fuks, Adam Adamandy Kochanski's approximations of pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739, 2011. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.

Crossrefs

Cf. A191642.

Programs

PARI

galpha(alpha, R, S) = {floor((alpha - floor(alpha))/(R - alpha*S));}
fuks() = { n = 29; default(realprecision, 200); alpha = sqrt(2); R = 3; S = 2; x = galpha(alpha, R, S); print1(x, ", "); for (i=1, n, R = R*(x+1) + floor(alpha); S = S*(x+1) + 1; x = galpha(alpha, R, S); print1(x, ", "););}
\\ Michel Marcus, Feb 07 2013

Formula

Definitions 1 and 2 of Fuks (2011) give formulas.

Extensions

More terms from Michel Marcus, Feb 07 2013