continued fraction of pi

The simple continued fraction for $\pi$ begins

 $\pi=3+\frac{1}{{7}+\frac{1}{{15}+\frac{1}{{1}+\,\cdots}}}$

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 ContinuedFractionOfPi 2013-03-22 16:35:57 2013-03-22 16:35:57 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Example msc 01A40 ContinuedFraction