A084407 Number of decimal places to which the n-th convergent of continued fraction expansion of Pi matches with the correct value.

0, 2, 4, 6, 9, 9, 9, 9, 11, 10, 12, 12, 14, 15, 15, 16, 17, 17, 18, 19, 21, 23, 24, 24, 25, 27, 29, 30, 30, 32, 33, 34, 37, 39, 40, 40, 41, 42, 44, 45, 45, 46, 47, 49, 50, 51, 51, 53, 54, 55, 55, 56, 56, 58, 59, 59, 60, 60, 61, 60, 62, 64, 63, 64, 65, 65, 67, 67, 68, 70, 69, 71

Offset: 1

Author

Lekraj Beedassy, Jun 24 2003

Keywords

• base
• nonn

Comments

The n-th convergent of the continued fraction expansion of Pi is A002485(n+1)/A002486(n+1).

Examples

			From _A.H.M. Smeets_, Jun 13 2018: (Start)
Pi = 3.141592653589...
n=1: 3/1 = 3.0... so a(1) = 0;
n=2: 22/7 = 3.142... so a(2) = 2;
n=3: 333/106 = 3.14150... so a(3) = 4;
n=4: 355/113 = 3.1415929... so a(4) = 6;
n=5: 103993/33102 = 3.1415926530... so a(5) = 9;
n=6: 104348/33215 = 3.1415926539... so a(6) = 9;
n=7: 208341/66317 = 3.1415926534... so a(7) = 9;
n=8: 312689/99532 = 3.1415926536... so a(8) = 9;
n=9: 833719/265381 = 3.141592653581... so a(9) = 11;
n=10: 1146408/364913 = 3.14159265359... so a(10) = 10. (End)
		

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..20000
• F. Richman, Continued fractions

Crossrefs

Cf. A000796.

Cf. A002485, A002486, A114526.

Cf. A086702, A100199, A240995.

Formula

Limit_{n -> oo} a(n)/n = 2*log(A086702)/log(10) = 2*A100199/log(10) = 2*A240995. - A.H.M. Smeets, Jun 13 2018

Extensions

More terms from Vladeta Jovovic, Jun 27 2003