A001203 -id:A001203 - cp's OEIS frontend

A033090 Indices of incrementally largest terms in the continued fraction for Pi.

1, 2, 3, 5, 308, 432, 28422, 156382, 267314, 453294, 11504931, 849955263, 2349980289, 3588031780, 8600404591, 15621034283

Offset: 1

Eric W. Weisstein, Bill Gosper

This sequence assumes nonstandard indexing of continued fraction terms as [a_1; a_2, a_3, ...]. If you use the actual offset from A001203, corresponding to [a_0; a_1, a_2, ...], you get instead 0, 1, 2, 4, 307, 431, 28421, ... Compare with A033092 versus A224849. - Jeppe Stig Nielsen, Dec 14 2019

Syed Fahad, 30 billion terms of the simple continued fraction of Pi
E. Fontich, C. Simó, and A. Vieiro, On the "hidden" harmonics associated to best approximants due to quasiperiodicity in splitting phenomena, Regular and Chaotic Dynamics (2018), Pleiades Publishing, Vol. 23, Issue 6, 638-653. Also PDF.
Eric Weisstein's World of Mathematics, Pi
Eric Weisstein's World of Mathematics, Pi Continued Fraction

Cf. A033089, A001203.

With[{s = ContinuedFraction[Pi, 2*10^7]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Jan 31 2020 *)

a(12) from Eric W. Weisstein, Dec 08 2010
a(13) from Eric W. Weisstein, Sep 16 2011
a(14) from Eric W. Weisstein, Sep 17 2011
a(15) from Eric W. Weisstein, Jul 18 2013
a(16) from Syed Fahad, Apr 27 2021

A154883 Distinct entries in continued fraction for Pi in the order of their appearance.

Original entry on oeis.org

3, 7, 15, 1, 292, 2, 14, 84, 13, 4, 6, 99, 5, 8, 12, 16, 161, 45, 22, 24, 10, 26, 42, 9, 57, 18, 19, 30, 28, 20, 120, 23, 21, 127, 29, 11, 48, 436, 58, 34, 44, 20776, 94, 55, 32, 50, 43, 72, 33, 27, 36, 106, 17, 141, 39, 125, 41, 37, 25, 47, 61, 376, 107, 31

Offset: 1

Lee Corbin (lcorbin(AT)rawbw.com), Jan 16 2009

This is presumably a permutation of the positive integers. The inverse permutation (or "index sequence") A322778 begins 4,6,1,10,13,11,2,14,... and gives the position in the continued fraction of Pi at which 1, 2, 3, 4, 5, 6, ... first appear. - Remark corrected by N. J. A. Sloane, Jan 04 2019

The name means that when a number not yet in this sequence appears in the continued fraction of Pi, then that number is added to the sequence. - T. D. Noe, May 06 2013

			Since the actual continued fraction for Pi is 3, 7, 15, 1, 292, 1, 1, 1, 2, ..., this sequence begins 3, 7, 15, 1, 292, 2, ...

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000, May 06 2013

Cf. A001203, A033089 (for records of main continued fraction), A322778 (inverse), A033090.

DeleteDuplicates[ContinuedFraction[Pi,1000]] (* Harvey P. Dale, May 06 2013 *)
t = {}; s = ContinuedFraction[Pi, 1000]; Do[If[! MemberQ[t, s[[n]]], AppendTo[t, s[[n]]]], {n, Length[s]}]; t (* T. D. Noe, May 06 2013 *)

\p 10000
v=contfrac(Pi); for(i=1,#v,for(j=1,i-1,if(v[i]==v[j],v[i]=0;break))); v=select(n->n,v) \\ Charles R Greathouse IV, May 06 2013

More terms from Harvey P. Dale, May 05 2013

A058291 Continued fraction for 2 Pi.

Original entry on oeis.org

6, 3, 1, 1, 7, 2, 146, 3, 6, 1, 1, 2, 7, 5, 5, 1, 4, 1, 2, 42, 5, 31, 1, 1, 1, 6, 2, 2, 4, 3, 12, 49, 1, 5, 1, 12, 1, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 16, 2, 1, 1, 15, 2, 3, 6, 3, 8, 18, 6, 1, 2, 1, 3, 2, 1, 2, 2, 1, 1, 2, 1, 8, 2, 33, 1, 80, 91, 1, 10, 1, 5, 1, 2, 2, 2, 1, 49, 2, 8, 2, 3, 5, 4, 2, 1, 1, 1

Offset: 0

Robert G. Wilson v, Dec 07 2000

A001203 is the continued fraction for Pi.

			6.283185307179586476925286766... = 6 + 1/(3 + 1/(1 + 1/(1 + 1/(7 + ...)))). - _Harry J. Smith_, May 31 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A019692 Decimal expansion. - Harry J. Smith, May 31 2009

Mathematica
```
ContinuedFraction[ 2Pi, 100 ]
```
PARI
```
contfrac(2*Pi)
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(2*Pi); for (n=0, 20000, write("b058291.txt", n, " ", x[n+1])); } \\ Harry J. Smith, May 31 2009

More terms from Jason Earls, Jul 24 2001

A373866 a(n) = maximum element in the continued fraction for Pi truncated to n decimal digits after the decimal point.

Original entry on oeis.org

3, 10, 7, 10, 14, 25, 84, 243, 288, 291, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 351, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292, 292

Offset: 0

Paolo Xausa, Jun 19 2024

			For n = 5, Pi truncated to 5 digits after the decimal point is 3.14159. The corresponding continued fraction is [3, 7, 15, 1, 25, 1, 7], whose maximum element is 25.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for continued fractions for constants

Cf. A000796, A001203, A081836 (analogous for phi), A081837 (analogous for e).

A373866[n_] := Max[ContinuedFraction[Floor[Pi*10^n]/10^n]];
Array[A373866, 100, 0]

A046956 Numbers k where tan(k) decreases monotonically to 0 (or cot(k) increases).

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 19, 22, 355, 104348, 312689, 1146408, 5419351, 85563208, 165707065, 411557987, 1480524883, 2549491779, 8717442233, 14885392687, 35938735828, 56992078969, 78045422110, 99098765251, 120152108392

Offset: 0

Olivier Gérard

nonn

From Jon E. Schoenfield, Aug 10 2006: (Start)
The approach described uses continued fractions containing an even number of terms of which all but the last term are fixed at the values those terms take in the continued fraction for Pi; the final term is initialized at 1 and incremented by 1 each time until it reaches the value taken by that term in the continued fraction for Pi. The semiconvergents and convergents thus obtained are increasingly accurate approximations for Pi, all of which approach Pi from values larger than Pi. Thus the angles whose sizes (in radians) are the numerators of those semiconvergents and convergents approach (from the positive side) integer multiples of Pi, so the tangents of those angles approach zero from positive values.
If we were to use the same approach but with continued fractions having an odd number of terms, i.e., [3] = 3/1; [3;7,i], i=1..15; [3;7,15,1,i], i=1..292; etc., then the semiconvergents and convergents obtained would likewise be increasingly accurate approximations for Pi, but they would approach Pi from values smaller than Pi, so the angles whose sizes (in radians) are the numerators of those semiconvergents and convergents would approach (from the negative side) integer multiples of Pi and thus the tangents of those angles would approach zero from negative values.
Terms after a(0) = 1 are the numerators of the fractions obtained by evaluating all those convergents and semiconvergents of the continued fraction for Pi (A001203) that, as written below, have an even number of partial quotients:
[3;i], i=1..7 (6 semiconvergents and 1 convergent)
[3;7,15,1]
[3;7,15,1,292,1]
[3;7,15,1,292,1,1,1]
[3;7,15,1,292,1,1,1,2,1]
[3;7,15,1,292,1,1,1,2,1,3,1]
[3;7,15,1,292,1,1,1,2,1,3,1,14,i], i=1..2
[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1]
[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,i], i=1..2
[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,i], i=1..2
[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,i], i=1..84, etc. (End)
See also A002485 which has a similar property (numerators of convergents to pi = numbers for which |tan a(n)| decreases to zero). - M. F. Hasler, Apr 01 2013

			a(1) is the numerator of [3;1] = 3 + 1/1 = 4/1
a(2) is the numerator of [3;2] = 3 + 1/2 = 7/2
...
a(7) is the numerator of [3;7] = 3 + 1/7 = 22/7
a(8) is the numerator of [3;7,15,1] = 3 + 1/(7 + 1/(15 + 1/1)) = 355/113
a(9) is the numerator of [3;7,15,1,292,1] = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + 1/1)))) = 104348/33215

Cf. A001203, A004112, A002485 (|tan a(n)|->0).

s = Tan[1]; Do[t = Tan[n]; If[t > 0 && t <= s, Print[n]; s = t], {n, 10^9}] (* Ryan Propper, Jul 27 2006 *)

e=2;for(n=1,1e9, tan(n)>0 && tan(n)M. F. Hasler, Apr 01 2013

More terms from Michel ten Voorde
2 more terms from Ryan Propper, Jul 27 2006
More terms from Jon E. Schoenfield, Aug 10 2006
Corrected by Don Reble, Nov 20 2006

A058651 Continued fraction for Pi + e.

Original entry on oeis.org

5, 1, 6, 7, 3, 21, 2, 1, 2, 2, 1, 1, 2, 3, 3, 2, 5, 2, 1, 1, 1, 1, 3, 1, 8, 4, 4, 1, 1, 1, 1, 8, 1, 4, 1, 5, 1, 1, 1, 2, 4, 3, 2, 1, 1, 2, 1, 10, 1, 4, 1, 2, 1, 12, 1, 8, 2, 7, 39, 365, 2, 15, 2, 25, 1, 2, 5, 3, 3, 9, 3, 1, 1, 9, 1, 1, 47, 1, 1, 18, 1, 1, 2, 6, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 4, 1, 6, 37

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Dec 26 2000

The question of the transcendence of the number Pi + e is still open.

			a(1) = 5 because Pi + e = 5.859874482048838473822930854632165381954416493075065395941912220031...
5.859874482048838473822930854... = 5 + 1/(1 + 1/(6 + 1/(7 + 1/(3 + ...)))). - _Harry J. Smith_, May 31 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A001203, A003417.

Cf. A059742 (decimal expansion).

PARI
```
\p 500; contfrac(Pi+exp(1))
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi+exp(1)); for (n=1, 20000, write("b058651.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 31 2009

More terms from Jason Earls, Jun 28 2001

Offset changed by Andrew Howroyd, Aug 04 2024

A137299 Square matrix read by antidiagonals: T(m,n) = m-th term in the continued fraction expansion of Pi^n.

Original entry on oeis.org

3, 9, 7, 31, 1, 15, 97, 159, 6, 1, 306, 2, 3, 1, 292, 961, 50, 2, 7, 2, 1, 3020, 2, 1, 3, 1, 47, 1, 9488, 3, 1, 4, 1, 13, 1, 1, 29809, 1, 2, 1, 60, 16539, 2, 8, 2, 93648, 10, 1, 2, 3, 1, 1, 1, 1, 1, 294204, 21, 14, 7, 3, 9, 4, 6, 3, 1, 3, 924269, 55, 15, 1, 1, 2, 1, 23, 7, 1, 2, 1

Offset: 1

M. F. Hasler, Mar 14 2008

The sequence was suggested by Leroy Quet.

			The matrix limited to order 10 is given by matrix(10,10,m,n,contfrac(Pi^n)[m]):
[   3   9   31    97   306   961  3020  9488 29809 93648]
[   7   1  159     2    50     2     3     1    10    21]
[  15   6    3     2     1     1     2     1    14    15]
[   1   1    7     3     4     1     2     7     1     1]
[ 292   2    1     1    60     3     3     1     9     4]
[   1  47   13 16539     1     9     2     1     3     2]
[   1   1    2     1     4     1    10     3     1     1]
[   1   8    1     6    23     5     4     1     5     3]
[   2   1    3     7     1     1     1     1     8     2]
[   1   1    1     6     2     3     1     1    16     1]

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)
J. S. Markovitch, Coincidence, data compression and Mach's concept of "economy of thought", APRI-PH-2004-12b, June 3 2004.

Cf. A001203, A001672, A138324.

A137299list[dmax_]:=With[{a=Array[ContinuedFraction[Pi^(dmax+1-#),#]&,dmax]},Array[Diagonal[a,#]&,dmax,1-dmax]];A137299list[10] (* Generates 10 antidiagonals *) (* Paolo Xausa, Nov 14 2023 *)

concat(vector(20,i,vector(i,j,contfrac(Pi^(i-j+1))[j])))

PARI
```
T(m,n)=contfrac(Pi^n)[m]
```

A177438 Continued fraction for Pi - sqrt(2).

Original entry on oeis.org

1, 1, 2, 1, 2, 77, 2, 1, 37, 1, 6, 1, 1, 1, 46, 3, 1, 1, 1, 1, 4, 2, 7, 1, 4, 1, 2, 1, 13, 1, 1, 1, 3, 2, 1, 1, 432, 1, 1, 1, 1, 3, 2, 10, 1, 1, 1, 18, 1, 1700, 1, 1, 5, 2, 9, 4, 4, 1, 1, 2, 1, 3, 27, 1, 1, 2, 1, 1, 1, 4, 3, 1, 2, 2, 5, 1, 32, 1, 11, 1, 2, 52, 10, 4, 1, 1, 10, 1, 1, 2, 23, 1, 3, 7, 12, 1

Offset: 0

Earl Bellinger (ebelling(AT)oswego.edu), May 08 2010

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A177437 (decimal expansion of Pi-sqrt(2)), A001203 (continued fraction for Pi), A040000 (continued fraction expansion of sqrt(2)).

SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(Pi(R) - Sqrt(2)); // G. C. Greubel, Sep 29 2018

with(numtheory): cfrac(Pi-(sqrt(2)),100,'quotients'); # Muniru A Asiru, Sep 29 2018

ContinuedFraction[Pi-Sqrt[2],100] (* Harvey P. Dale, Nov 06 2011 *)

default(realprecision, 100); contfrac(Pi - sqrt(2)) \\ G. C. Greubel, Sep 29 2018

Edited and extended by Klaus Brockhaus, May 09 2010

A225802 Position of first occurrence of n in continued fraction for Pi, or -1 if n never occurs.

Original entry on oeis.org

3, 8, 0, 29, 39, 31, 1, 43, 129, 99, 275, 54, 27, 12, 2, 77, 646, 136, 139, 179, 213, 82, 202, 90, 790, 111, 573, 174, 242, 146, 877, 454, 530, 420, 1007, 593, 783, 3040, 720, 1871, 753, 118, 491, 428, 80, 3199, 824, 282, 3026, 464, 1436, 3383, 1546, 1863, 445, 1017

Offset: 1

Eric W. Weisstein, Jul 27 2013

nonn

Correctly indexed version of A032523.

All positive integers <= 49003 occur in the first 15000000000 terms of the c.f. (the first that do not are 49004, 50471, 53486, 56315, 58255, ...) - Eric W. Weisstein, Jul 27 2013

			The continued fraction of Pi is [a_0; a_1, a_2, ...] = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, ...], so
a(1) = 3 (1 first occurs at term a_3);
a(2) = 8 (2 first occurs at term a_8);
a(3) = 0 (3 first occurs at term a_0).

Eric W. Weisstein, Table of n, a(n) for n = 1..49003
Eric Weisstein's World of Mathematics, Pi Continued Fraction

Cf. A032523 (= a(n) + 1).

Cf. A001203 (continued fraction of Pi).

a(n) = A032523(n) - 1.

"Escape clause" added to definition by Jianing Song, Apr 06 2019

A273128 Continued logarithm expansion of Pi.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 3, 0, 3, 0, 2, 0, 0, 2, 5, 0, 1, 2, 0, 1, 2, 4, 3, 0, 0, 1, 2, 0, 0, 1, 7, 3, 2, 3, 0, 1, 0, 0, 0, 0, 3, 2, 3, 1, 3, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 0, 1, 2, 1, 0, 3, 4, 0, 0, 0, 3, 0, 0, 0, 1, 1, 1, 2, 2, 0, 3, 0, 0, 1, 0, 2, 1, 3, 0, 0, 2, 0

Offset: 1

Jeffrey Shallit, May 16 2016

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..10000
Jon Borwein, Neil Calkin, Scott Lindstrom, and Andrew Mattingly, Continued logarithms and associated continued fractions, preprint, 2016.

Cf. A000796, A001203.

Cf. A273127 (similar for e).

default(realprecision, 10^5); p=Pi;
lista(nn) = {my(c); for(n=1, nn, print1(c=logint(p\1, 2), ", "); p=1/(p/2^c-1)); } \\ Jinyuan Wang, Feb 27 2020

cp's OEIS Frontend

A033090 Indices of incrementally largest terms in the continued fraction for Pi.

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A154883 Distinct entries in continued fraction for Pi in the order of their appearance.

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Mathematica

PARI

Extensions

A058291 Continued fraction for 2 Pi.

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Mathematica

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PARI

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A373866 a(n) = maximum element in the continued fraction for Pi truncated to n decimal digits after the decimal point.

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Mathematica

A046956 Numbers k where tan(k) decreases monotonically to 0 (or cot(k) increases).

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Programs

Mathematica

PARI

Extensions

A058651 Continued fraction for Pi + e.

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PARI

PARI

Extensions

A137299 Square matrix read by antidiagonals: T(m,n) = m-th term in the continued fraction expansion of Pi^n.

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