by neufer » Sun Mar 15, 2020 4:06 am
heehaw wrote: ↑Sat Mar 14, 2020 10:17 pm
The string 0123456789876543210 occurs infinitely many times in pi.
More interesting, perhaps, is the fact that
every number occurs infinitely many times in the continued fraction coefficients of pi:
3;7,15,1,292,1,1,1,2,1,3,1,...
https://en.wikipedia.org/wiki/Continued_fraction wrote:
<<In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers a
i are called the coefficients or terms of the continued fraction.>>
Continued fraction coefficients of irrational numbers:
ϕ = [1;1,1,1,1,1,1,1,1,1,1,1,...] (sequence A000012 in the OEIS). The golden ratio, the irrational number that is the "most difficult" to approximate rationally. See: A property of the golden ratio φ.
√19 = [4;2,1,3,1,2,8,2,1,3,1,2,8,...] (sequence A010124 in the OEIS). The pattern repeats indefinitely with a period of 6.
e = [2;1,2,1,1,4,1,1,6,1,1,8,...] (sequence A003417 in the OEIS). The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.
π = [
3;7,15,1,292,1,1,1,2,1,3,1,...] (sequence A001203 in the OEIS).
No pattern has ever been found in this representation.
https://oeis.org/A001203 wrote:
Simple continued fraction expansion of Pi:
3, 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, ...
https://oeis.org/A032523 wrote:
Index of first occurrence of n as a [coefficient] in the continued fraction for Pi:
4, 9, 1, 30, 40, 32, 2, 44, 130, 100, 276, 55, 28, 13, 3, 78, 647, 137, 140, 180, 214, 83, 203, 91, 791, 112, 574, 175, 243, 147, 878, 455, 531, 421, 1008, 594, 784, 3041, 721, 1872, 754, 119, 492, 429, 81, 3200, 825, 283, 3027, 465, 1437, 3384, 1547, 1864, 446, ...
[quote=heehaw post_id=300349 time=1584224237]
The string 0123456789876543210 occurs infinitely many times in pi.[/quote]
More interesting, perhaps, is the fact that
[b][color=#0000FF]every number occurs infinitely many times in the continued fraction coefficients of pi:[/color][/b]
[c][b][color=#0000FF]3;7,15,1,292,1,1,1,2,1,3,1,...[/color][/b][/c]
[quote=https://en.wikipedia.org/wiki/Continued_fraction]
[float=left][img3=""]https://wikimedia.org/api/rest_v1/media/math/render/svg/247535cef4b9b94eabeb16908cf72436cd01d0c9[/img3][/float]
<<In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers a[sub]i[/sub] are called the coefficients or terms of the continued fraction.>>
Continued fraction coefficients of irrational numbers:
ϕ = [1;1,1,1,1,1,1,1,1,1,1,1,...] (sequence A000012 in the OEIS). The golden ratio, the irrational number that is the "most difficult" to approximate rationally. See: A property of the golden ratio φ.
√19 = [4;2,1,3,1,2,8,2,1,3,1,2,8,...] (sequence A010124 in the OEIS). The pattern repeats indefinitely with a period of 6.
e = [2;1,2,1,1,4,1,1,6,1,1,8,...] (sequence A003417 in the OEIS). The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.
π = [[b][color=#0000FF]3;7,15,1,292,1,1,1,2,1,3,1,...[/color][/b]] (sequence A001203 in the OEIS). [b][u][color=#0000FF]No pattern has ever been found in this representation.[/color][/u][/b][/quote][quote=https://oeis.org/A001203]
Simple continued fraction expansion of Pi:
[b][color=#0000FF]3, 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, ...[/color][/b][/quote][quote=https://oeis.org/A032523]
[b][color=#FF0000]Index of first occurrence of n[/color][/b] as a [coefficient] in the continued fraction for Pi:
[b][color=#FF0000]4, 9, 1, 30, 40, 32, 2, 44, 130, 100, 276, 55, 28, 13, 3, 78, 647, 137, 140, 180, 214, 83, 203, 91, 791, 112, 574, 175, 243, 147, 878, 455, 531, 421, 1008, 594, 784, 3041, 721, 1872, 754, 119, 492, 429, 81, 3200, 825, 283, 3027, 465, 1437, 3384, 1547, 1864, 446, ...[/color][/b]
[quote=http://chesswanks.com/pxp/cfpi.html]
[b][color=#0000FF]Of course, every integer [in the continued fraction for Pi] is not just expected to show up once, but an unlimited number of times.[/color][/b][/quote]