continued fraction of pi

continued fraction of pi

The

simple continued fraction

for

π

begins

π

=

3

+

1

7

+

1

15

+

1

1

+

⋯

and continues with the

numerators

292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, … listed in A001203 of Sloane’s OEIS.

Here are a few more terms: 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 16, 1, 161, 45, 1, 22, 1, 2, 2, 1, 4, 1, 2, 24, 1, 2, 1, 3, 1, 2, 1, 1, 10, 2, 5, 4, 1, 2, 2, 8, 1, 5, 2, 2, 26, 1, 4, 1, 1, 8, 2, 42, 2, 1, 7, 3, 3, 1, 1, 7, 2, 4, 9, 7, 2, 3, 1, 57, 1, 18, 1, 9, 19, 1, 2, 18, 1, 3, 7, 30, 1, 1, 1, 3, 3, 3, 1, 2, 8, 1, 1, 2, 1, 15, 1, 2, 13, 1, 2, 1, 4, 1, 12, 1, 1, 3, 3, 28, 1, 10, 3, 2, 20, 1, 1, 1, 1, 4, 1, 1, 1, 5, 3, 2, 1, 6, 1, 4, 1, 120, 2, 1, 1, 3, 1, 23, 1, 15, 1, 3, 7, 1, 16, 1, 2, 1, 21, 2, 1, 1, 2, 9, 1, 6, 4, 127, 14, 5, 1, 3, 13, 7, 9, 1, 1, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 29, 3, 1, 1, 2, 2, 1, 3, 1, 1, 1, 3, 1, 1, 10, 3, 1, 3, 1, 2, 1, 12, 1, 4, 1, 1, 1, 1, 7, 1, 1, 2, 1, 11, 3, 1, 7, 1, 4, 1, 48, 16, 1, 4, 5, 2, 1, 1, 4, 3, 1, 2, 3, 1, 2, 2, 1, 2, 5, 20, 1, 1, 5, 4, 1, 436, 8, 1, 2, 2, 1, 1, 1, 1, 1, 5, 1, 2, 1, 3, 6, 11, 4, 3, 1, 1, 1, 2, 5, 4, 6, 9, 1, 5, 1, 5, 15, 1, 11, 24, 4, 4, 5, 2, 1, 4, 1, 6, 1, 1, 1, 4, 3, 2, 2, 1, 1, 2, 1, 58, 5, 1, 2, 1, 2, 1, 1, 2, 2, 7, 1, 15, 1, 4, 8, 1, 1, 4, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 9, 1, 4, 3, 15, 1, 2,
1, 13, 1, 1, 1, 3, 24, 1, 2, 4, 10, 5, 12, 3, 3, 21, 1, 2, 1, 34, 1, 1, 1, 4, 15, 1, 4, 44, 1, 4, 20776, 1, 1, 1, 1, 1, 1, 1, 23, 1, 7, 2, 1, 94, 55, 1, 1, 2, 1, 1, 3, 1, 1, 32, 5, 1, 14, 1, 1, 1, 1, 1, 3, 50, 2, 16, 5, 1, 2, 1, 4, 6, 3, 1, 3, 3, 1, 2, 2, 2, 5, 2, 2, 2, 28, 1, 1, 13, 1, 5, 43, 1, 4, 3, 5, 3, 1, 4, 1, 1, 2, 2, 1, 1, 19, 2, 7, 1, 72, 3, 1, 2, 3, 7, 11, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 33, 7, 19, 1, 19, 3, 1, 4, 1, 1, 1, 1, 2, 3, 1, 3, 2, 2, 2, 2, 4, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 11, 1, 1, 2, 1, 2, 1, 2, 2, 1, 7, 2, 27, 1, 1, 6, 2, 1, 9, 6, 26, 1, 1, 3, 2, 1, 1, 1, 1, 1, 15, 1, 36, 4, 2, 2, 1, 22, 2, 1,
106, 2, 2, 1, 3, 1, 12, 10, 7, 1, 2, 1, 1, 1, 1, 8, 2, 4, 5, 3, 2, 1, 4, 23, 1, 18, 2, 10, 3, 1, 6, 6, 13, 8, 6, 2, 2, 2, 2, 1, 1, 1, 3, 1, 7, 17, 1, 1, 1, 2, 5, 5, 1, 1, 2, 11, 1, 6, 1, 6, 1, 29, 4, 29, 3, 5, 3, 1, 141, 1, 2, 7, 7, 2, 2, 7, 1, 1, 7, 1, 7, 1, 2, 4, 1, 1, 1, 30, 1, 12, 4, 18, 10, 2, 8, 1, 2, 2, 2, 4, 13, 1, 5, 4, 1, 6, 1, 1, 11, 2, 4, 2, 1, 1, 3, 3, 12, 1, 1, 39, 5, 1, 1, 16, 125, 1, 4, 1, 2, 1, 19, 1, 4, 1, 1, 2, 1, 4, 1, 10, 1, 4, 2, 1, 1, 1, 5, 10, 4, 14, 1, 13, 41, 1, 4, 1, 8, 1, 1, 2, 1, 3, 1, 6, 1, 3, 2, 2, 2, 1, 4, 1, 14, 1, 2, 8, 1, 8, 3, 3, 3, 1, 37, 4, 2, 4, 1, 3, 4, 25, 4, 27, 2, 7, 1, 1, 2, 6, 1, 1,
1, 12, 1, 2, 2, 2, 13, 12, 1, 3, 1, 6, 1, 1, 33, 1, 5, 3, 1, 5, 15, 8, 8, 47, 1, 3, 2, 12, 2, 12, 1, 12, 1, 2, 5, 3, 1, 1, 1, 1, 2, 3, 5, 4, 2, 1, 1, 5, 1, 9, 14, 1, 1, 3, 2, 1, 9, 3, 22, 13, 1, 1, 3, 20, 1, 1, 61, 1, 376, 2, 107, 1, 10, 3, 2, 2, 31, 1, 2, 10, 2, 2, 62, 2, 2, 7, 4, 5, 6, 1, 1, 1, 1, 2, 8, 2, 73, 3, 5, 42, 1, 3, 2, 1, 1, 59, 6, 1, 1, 1, 5, 1, 6, 1, 2, 6, 1, 1, 1, 1, 3, 2, 1, 3, 1, 8, 1, 4, 2, 5, 4, 7, 1, 4, 2, 2, 6, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 5, 1, 2, 1, 1, 10, 1, 6, 1, 129, 1, 4, 65, 2, 4, 4, 3, 2, 3, 1, 1, 5, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 1, 2, 4, 2, 1, 2, 27, 6, 2, 1, 193, 1, 3, 9, 1, 3, 35, 2, 1, 8, 1, 1, 1, 1, 9, 3, 56, 1, 6, 6, 2, 8, 1, 8, 1, 2, 3, 6, 3, 1, 3, 1, 1, 1, 2, 13, 1, 1, 1, 1, 13, 2, 1, 3, 1, 3, 15, 2, 1, 1, 2, 4, 1, 4, 5, 2, 2, 1, 2, 1, 6, 1, 4, 12, 1, 1, 1, 1, 13, 1, 3, 4, 1, 1, 1, 2, 9, 1, 7, 1, 1, 1, 1, 4, 1, 3, 4, 1, 1, 4, 3, 1, 39, 2, 1, 1, 1, 1, 1, 4, 7, 2, 2, 2, 1, 1, 1, 1, 2, 114, 12, 4, 1, 3, 2, 1, 19, 1, 1, 2, 1, 1, 3, 4, 1, 60, 3, 72, 2, 1, 1, 1, 50, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 7, 3, 1, 2, 1, 5, 1, 1, 1, 2, 6, 2, 21, 2, 6, 1, 6, 1, 1, 2, 1, 7, 1, 8, 1, 1, 5, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 11, 2, 4, 10, 2, 1, 1, 13, 1, 1, 7, 15, 1, 1, 1, 2, 3, 15, 8, 8, 2, 1, 13, 3, 5, 1, 2, 1, 6, 1, 10, 123, 3, 1, 4, 59, 4, 156, 88, 1, 5, 4, 1, 3, 1, 4, 2, 9, 1, 7, 4, 2, 1, 2, 3, 2, 1, 2, 11, 1, 13, 7, 7, 1, 63, 37, 12, 86, 1, 1, 1, 1, 2, 2, 4, 2, 18, 1, 1, 1, 41, 2, 1, 1, 12, 1, 2, 1, 1, 2, 10, 1, 1, 1, 5, 1, 1, 3, 1, 7, 5, 1, 9, 1, 2, 2, 7, 1, 1, 5, 2, 1, 3, 3, 5, 2, 1, 11, 3, 1, 3, 2, 1, 1, 2, 1, 14, 5, 2, 2, 1, 1, 1, 1, 3, 1, 3, 3, 2, 2, 1, 3, 2, 1, 2, 1, 4, 1, 14, 1, 1, 58, 7, 1, 2, 1, 1, 5, 1, 2, 1, 5, 18, 1, 4, 3, 1, 1, 1, 4, 1, 1, 2, 5, 1, 148, 1, 9, 2, 1, 2, 1, 5, 4, 93, 1, 1, 2, 4, 1, 2, 73, 1, 1, 3, 1, 1, 1, 1, 2, 1, 34, 1, 5, 6, 1, 2, 1, 3, 4, 1, 16, 28, 17, 2, 5, 5, 26, …

Most

computer algebra systems

provide relatively simple commands for obtaining as many terms as one wants. In

Mathematica

, for example, one can issue the command

ContinuedFraction[Pi, 1000]

, replacing 1000 with whichever number one prefers. In this particular case, it only takes Mathematica a couple of seconds to give a thousand terms.

Title

continued fraction of pi

Canonical name

ContinuedFractionOfPi

Date of creation

2013-03-22 16:35:57

Last modified on

2013-03-22 16:35:57

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 01A40

Related topic

ContinuedFraction

Generated on Thu Feb 8 19:08:39 2018 by

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