A001203 -id:A001203 - cp's OEIS frontend

A276459 Nested radical expansion of Pi: Pi = sqrt(a(1) + sqrt(a(2) + sqrt(a(3) + sqrt(a(4) + ...)))), with a(1) = 7 and 2 <= a(n) <= 6 for n>1.

7, 6, 2, 6, 6, 5, 5, 2, 4, 6, 3, 4, 2, 4, 6, 3, 6, 3, 3, 5, 4, 3, 6, 3, 3, 3, 4, 3, 6, 6, 4, 3, 3, 4, 5, 5, 2, 6, 2, 5, 4, 3, 4, 6, 6, 2, 3, 5, 2, 3, 5, 4, 2, 3, 2, 4, 2, 6, 4, 6, 3, 3, 4, 3, 4, 6, 3, 4, 6, 5, 2, 2, 2, 3, 4, 5, 5, 5, 2, 4, 3, 6, 4, 3, 6, 3, 2, 6, 2, 4, 5, 6, 2, 3, 2, 5, 2, 3, 2, 3, 3, 5, 4, 4, 6, 4, 2, 4, 5, 4, 6, 5, 3

Offset: 1

Yuriy Sibirmovsky, Sep 03 2016

nonn

Similar to Bolyai expansion. Uses the fact that for 0

			Pi^2=7+2+p1, thus a(1)=7;
(2+p1)^2=6+2+p2, thus a(2)=6;
(2+p2)^2=2+2+p3, thus a(3)=2; ... 0<pn<1.

Yuriy Sibirmovsky, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Bolyai Expansion.

Cf. A000796 (digits), A001203 (continued fraction).

Nm=100;
A=Table[1,{j,1,Nm}];
V=Table[1,{j,1,Nm}];
P=Pi;
p0=P;
Do[p1=Floor[p0^2]-2;
A[[j]]=p1;
p0=N[2+p0^2-Floor[p0^2],300],{j,1,Nm}];
Do[v0=Sqrt[A[[n]]];
Do[v1=A[[n-j]]+v0;
v0=Sqrt[v1],{j,1,n-1}];
V[[n]]=v0,{n,1,Nm}];
A

A280135 Negative continued fraction of Pi (also called negative continued fraction expansion of Pi).

Original entry on oeis.org

4, 2, 2, 2, 2, 2, 2, 17, 294, 3, 4, 5, 16, 2, 3, 4, 2, 4, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Randy L. Ekl, Dec 26 2016

nonn

Appears that these terms are related to continued fraction of Pi through simple transforms; original continued fraction terms X,1 -> negative continued fraction term X+2 (e.g., 15,1->17, and 292,1->294); other transforms are to be determined.

			Pi = 4 - (1 / (2 - (1 / (2 - (1 / ...))))).

Leonard Eugene Dickson, History of the Theory of Numbers, page 379.

Cf. A001203 (continued fraction of Pi).
Cf. A133593 (exact continued fraction of Pi).
Cf. A280136 (negative continued fraction of e).

\p10000; p=Pi;for(i=1,300,print(i," ",ceil(p)); p=ceil(p)-p;p=1/p )

A292106 Term-by-term products of continued fraction expansion of e and Pi.

Original entry on oeis.org

6, 7, 30, 1, 292, 4, 1, 1, 12, 1, 3, 8, 14, 2, 10, 1, 2, 24, 2, 2, 14, 84, 2, 16, 1, 15, 54, 13, 1, 80, 2, 6, 132, 99, 1, 48, 2, 6, 78, 5, 1, 28, 6, 8, 30, 7, 1, 64, 3, 7, 34, 2, 1, 36, 12, 1, 38, 1, 3, 40, 1, 8, 42, 1, 2, 44, 6, 1, 46, 5, 2, 96, 3, 1, 100, 4

Offset: 0

N. J. A. Sloane, Sep 14 2017

nonn

Paolo Xausa, Table of n, a(n) for n = 0..9999

Cf. A001203, A003417, A292107.

With[{nmax=100},ContinuedFraction[E,nmax]ContinuedFraction[Pi,nmax]] (* Paolo Xausa, Oct 22 2023 *)

a(n) = A003417(n) * A001203(n).

Offset changed by Andrew Howroyd, Aug 07 2024

A322778 Position at which n first appears in the continued fraction expansion of Pi.

Original entry on oeis.org

4, 6, 1, 10, 13, 11, 2, 14, 24, 21, 36, 15, 9, 7, 3, 16, 53, 26, 27, 30, 33, 19, 32, 20, 59, 22, 50, 29, 35, 28, 64, 45, 49, 40, 71, 51, 58, 107, 55, 93, 57, 23, 47, 41, 18, 111, 60, 37, 106, 46, 82, 119, 86, 91, 44, 72, 25, 39, 67, 74, 61, 65, 78, 154, 69, 204, 89, 169, 167, 105, 198

Offset: 1

N. J. A. Sloane, Jan 04 2019

nonn

This is presumably a permutation of the positive integers, and is the inverse permutation to A154883.

R. J. Mathar, Table of n, a(n) for n = 1..292, Jan 11 2019

Cf. A001203, A154883.

A048292 1-digit terms in the continued fraction for Pi.

Original entry on oeis.org

3, 7, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 1, 1, 3, 1, 4, 2, 6, 6, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 1, 1, 1, 2, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 2, 5, 4, 1, 2, 2, 8, 1, 5, 2, 2, 1, 4, 1, 1, 8

Offset: 1

Eric W. Weisstein

			A001203 begins (3, 7, 15, 1, 292, 1, ...) so sequence begins (3, 7, 1, 1, ...).

Eric Weisstein's World of Mathematics, Pi Continued Fraction.

Cf. A001203, A048293.

Select[ContinuedFraction[Pi,200],#<10&] (* Harvey P. Dale, Jan 29 2023 *)

A048293 Positions of 1-digit terms in the continued fraction for Pi (3 is at position 0).

Original entry on oeis.org

0, 1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 81

Offset: 1

Eric W. Weisstein

Eric Weisstein's World of Mathematics, Pi Continued Fraction

Cf. A001203, A048292.

A091657 Length of the smallest prefix of the continued fraction expansion for Pi that includes each of 1..n.

Original entry on oeis.org

4, 9, 9, 30, 40, 40, 40, 44, 130, 130, 276, 276, 276, 276, 276, 276, 647, 647, 647, 647, 647, 647, 647, 647, 791, 791, 791, 791, 791, 791, 878, 878, 878, 878, 1008, 1008, 1008, 3041, 3041, 3041, 3041, 3041, 3041, 3041, 3041, 3200, 3200, 3200, 3200, 3200, 3200

Offset: 1

Robert G. Wilson v, Jan 26 2004

nonn

Cf. A001203, A032523.

f[n_] := Block[{k = 1}, While[ StringPosition[ ToString[ Union[ ContinuedFraction[Pi, k]]], StringDrop[ ToString[ Table[i, {i, n}]], -1]] == {}, k++ ]; k]; Table[ f[n], {n, 1, 51}]

a(n) = max(A032523(n), a(n-1)) for n > 1. - Andrew Howroyd, Aug 05 2024

Name clarified by Andrew Howroyd, Aug 05 2024

A099643 Continued fraction for Pi + 1/Pi.

Original entry on oeis.org

3, 2, 5, 1, 2, 1, 3, 2, 1, 2, 1, 13, 897, 2, 8, 1, 2, 1, 1, 15, 2, 3, 2, 1, 1, 2, 1, 11, 2, 1117, 1, 12, 2, 1, 1, 2, 1, 18, 4, 187, 22, 1, 3, 3, 3, 14, 1, 2, 3, 1, 78, 3, 1, 1, 65, 3, 1, 4, 2, 2, 2, 1, 3, 16, 2, 5, 5, 2, 1, 4, 1, 3, 883, 1, 1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 15, 1, 2, 5, 1, 1, 2, 1, 9, 16, 1

Offset: 0

Labos Elemer, Nov 02 2004

Pi and 1/Pi have separately x and 1-shifted-x continued fraction coefficient series. This expansion apparently does not display connection with those of added terms.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001203, A098801 (decimal expansion).

Mathematica
```
ContinuedFraction[Pi+1/Pi, 128]
```

contfrac(Pi + 1/Pi) \\ Amiram Eldar, Mar 08 2025

Offset changed by Andrew Howroyd, Aug 04 2024

A137995 Nearest integer to 1/frac(Pi^A137994(n)), where frac(x) = x - floor(x).

Original entry on oeis.org

7, 159, 270, 308, 745, 758, 796, 1080, 1227, 7805, 13876, 62099, 70718, 86902, 154756

Offset: 1

M. F. Hasler, inspired by Leroy Quet, Apr 05 2008

Sequence A137994 could be defined as "least positive integer such that this one (without rounding) is increasing".

The term a(1)=7 is not surprising (3 + 1/7 = 3.14...) but it comes as a funny surprise that the next term, a(2)=159, matches the next 3 digits of Pi and a(3) just differs by 5 from the next 3 digits!

Cf. A137994, A001203, A001672, A137299, A138324.

default(realprecision,10^4); f=1; for(i=1,10^9, frac(Pi^i)

a(7) inserted and a(11)-a(15) added by Amiram Eldar, Jun 28 2025

A159824 Continued fraction for Pi^Pi (cf. A073233).

Original entry on oeis.org

36, 2, 6, 9, 2, 1, 2, 5, 1, 1, 6, 2, 1, 291, 1, 38, 50, 1, 2, 5, 4, 1, 2, 2, 1, 5, 1, 4, 13, 2, 1, 4, 3, 3, 1, 2, 25, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 1, 43, 1, 2, 7, 3, 1, 1, 1, 2, 4, 2, 1, 1, 3, 1, 3, 3, 2, 2, 16, 3, 5, 2, 1, 5, 2, 1, 10, 1, 1, 3, 1, 13, 1, 1, 3, 1, 10, 4, 1, 1, 1, 38, 1, 2, 2, 1, 1, 3

Offset: 0

Harry J. Smith, Apr 30 2009

nonn

			36.4621596072079117709908260... = 36 + 1/(2 + 1/(6 + 1/(9 + 1/(2 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A001076, A001203, A001204, A002945, A010767, A011002, A011003, A073233, A073238, A179613, A179615, A179616, A179617.

ContinuedFraction[Pi^Pi,200] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2010 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^Pi); for (n=1, 20001, write("b159824.txt", n-1, " ", x[n])); }

Edited by N. J. A. Sloane, Jul 22 2010

Previous Showing 31-40 of 51 results. Next

cp's OEIS Frontend

A276459 Nested radical expansion of Pi: Pi = sqrt(a(1) + sqrt(a(2) + sqrt(a(3) + sqrt(a(4) + ...)))), with a(1) = 7 and 2 <= a(n) <= 6 for n>1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A280135 Negative continued fraction of Pi (also called negative continued fraction expansion of Pi).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

A292106 Term-by-term products of continued fraction expansion of e and Pi.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A322778 Position at which n first appears in the continued fraction expansion of Pi.

Views

Author

Keywords

Comments

Links

Crossrefs

A048292 1-digit terms in the continued fraction for Pi.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A048293 Positions of 1-digit terms in the continued fraction for Pi (3 is at position 0).

Views

Author

Keywords

Links

Crossrefs

A091657 Length of the smallest prefix of the continued fraction expansion for Pi that includes each of 1..n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A099643 Continued fraction for Pi + 1/Pi.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A137995 Nearest integer to 1/frac(Pi^A137994(n)), where frac(x) = x - floor(x).

Views

Author

Keywords

Comments

Crossrefs

Programs