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

1,2

Comments

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

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

F. Richman, Continued fractions

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

Example

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)

Crossrefs

Cf. A000796.

Cf. A002485, A002486, A114526.

Cf. A086702, A100199, A240995.

Sequence in context: A176461 A255249 A330394 * A114526 A333387 A178126

Adjacent sequences: A084404 A084405 A084406 * A084408 A084409 A084410

Keyword

base,nonn

Author

Lekraj Beedassy, Jun 24 2003

Extensions

More terms from Vladeta Jovovic, Jun 27 2003

Status

approved