A000796 -id:A000796 - cp's OEIS frontend

A089491 Decimal expansion of Buffon's constant 3/Pi.

9, 5, 4, 9, 2, 9, 6, 5, 8, 5, 5, 1, 3, 7, 2, 0, 1, 4, 6, 1, 3, 3, 0, 2, 5, 8, 0, 2, 3, 5, 0, 8, 6, 1, 7, 2, 2, 0, 6, 7, 5, 7, 8, 7, 4, 4, 4, 2, 7, 3, 8, 6, 9, 2, 4, 8, 6, 0, 0, 4, 0, 6, 4, 3, 5, 3, 3, 8, 0, 7, 8, 5, 8, 0, 5, 3, 5, 9, 2, 1, 0, 5, 4, 0, 6, 8, 2, 8, 1, 6, 5, 9, 7, 5, 1, 8, 5, 1, 5, 7, 3, 6, 4, 3, 7

Offset: 0

Robert G. Wilson v, Nov 04 2003

Whereas 2/Pi (A060294) is the probability that a needle will land on one of many parallel lines, this is the probability that a needle will land on one of many lines making up a grid.
The probability that the boundary of an equilateral triangle will intersect one of the parallel lines if the triangle edge length l (almost) equals the distance d between each pair of lines. This follows directly from the Weisstein/MathWorld Buffon's Needle Problem link's statement P=p/(Pi*d), where P is the probability of intersection with any convex polygon's boundary if the generalized diameter of that polygon is less than d and p is the perimeter of the polygon. (Take d=l, then p=3d.) - Rick L. Shepherd, Jan 11 2006
Related grid problems are discussed in the Weisstein/MathWorld Buffon-Laplace Needle Problem link. - Rick L. Shepherd, Jan 11 2006
The area of a regular dodecagon circumscribed in a unit-area circle. - Amiram Eldar, Nov 05 2020
From Bernard Schott, Apr 19 2022: (Start)
For any non-obtuse triangle ABC (see Mitrinović and Oppenheim links):
(a/A + b/B + c/C)/(a+b+c) >= 3/Pi,
(a^2/A + b^2/B + c^2/C)/(a^2+b^2+c^2) <= 3/Pi,
where (A,B,C) are the angles (measured in radians) and (a,b,c) the side lengths of this triangle.
Equality stands iff triangle ABC is equilateral. (End)

			3/Pi = 0.95492965855137201461330258023508617220675787444273869248600...

Joe Portney, Portney's Ponderables, Litton Systems, Inc., Appendix 2, 'Buffon's Needle' by Lawrence R. Weill, 200, pp. 135-138.

Harry Khamis, Buffon's Needle Problem.
D. S. Mitrinović, J. E. Pečarić, and V. Volenec, Recent Advances In Geometric Inequalities, Kluwer Academic Publishers, 1989, Inequalities 4.11, p. 170.
A. Oppenheim, Problem E 2649, American Mathematical Monthly, 84 (1977), p. 294.
Kevin Peterson, A Problem in Geometric Probability: Buffon's Needle Problem.
George Reese, Buffon's Needle, An Analysis and Simulation.
Shodor Education Foundation, Inc., Buffon's needle.
Washington and Lee University, Problem 18: Buffon's Needle Again. [Broken link]
Eric Weisstein's World of Mathematics, Buffon's needle problem.
Eric Weisstein's World of Mathematics, Buffon-Laplace needle problem.
Eric Weisstein's World of Mathematics, Generalized Diameter.
Index entries for transcendental numbers

Cf. A000796 (Pi), A060294 (2/Pi).

Mathematica
```
RealDigits[ N[ 3/Pi, 111]][[1]]
```
PARI
```
3/Pi \\ Michel Marcus, Nov 05 2020
```

Equals sinc(Pi/6). - Peter Luschny, Oct 04 2019
From Amiram Eldar, Aug 20 2020: (Start)
Equals Product{k>=1} cos(Pi/(6*2^k)).
Equals Product{k>=1} (1 - 1/(6*k)^2). (End)

A004608 Expansion of Pi in base 9.

Original entry on oeis.org

3, 1, 2, 4, 1, 8, 8, 1, 2, 4, 0, 7, 4, 4, 2, 7, 8, 8, 6, 4, 5, 1, 7, 7, 7, 6, 1, 7, 3, 1, 0, 3, 5, 8, 2, 8, 5, 1, 6, 5, 4, 5, 3, 5, 3, 4, 6, 2, 6, 5, 2, 3, 0, 1, 1, 2, 6, 3, 2, 1, 4, 5, 0, 2, 8, 3, 8, 6, 4, 0, 3, 4, 3, 5, 4, 1, 6, 3, 3, 0, 3, 0, 8, 6, 7, 8, 1, 3, 2, 7, 8, 7, 1, 5, 8, 8, 5, 3, 6, 8, 1, 3, 6, 5, 3

Offset: 1

N. J. A. Sloane

			3.12418812407442788645177761731035828516...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steve Pagliarulo, Stu's pi page: base 9 (23 pages) [Capture on Wayback Machine]
Steve Pagliarulo, Stu's pi page: base 9 (recovered link)

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), this sequence (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).

Cf. A007514.

RealDigits[Pi, 9, 105][[1]]
Table[ResourceFunction["NthDigit"][Pi, n, 9], {n, 1, 105}] (* Joan Ludevid, Oct 09 2022 easy to compute a(10000000)=5 with this function; requires Mathematica 12.0+ *)

More terms from Robert G. Wilson v, Oct 20 2002

A037001 Positions of the digit '2' in the decimal expansion of Pi (where positions 0, 1, 2,... refer to the digits 3, 1, 4,...).

Original entry on oeis.org

6, 16, 21, 28, 33, 53, 63, 73, 76, 83, 89, 93, 102, 112, 114, 135, 136, 140, 149, 160, 165, 173, 185, 186, 203, 221, 229, 241, 244, 260, 275, 280, 289, 292, 298, 302, 326, 329, 333, 335, 337, 354, 374, 380, 406, 423, 435, 456, 462, 477, 479, 484, 485, 500

Offset: 1

Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br)

The first few primes in this sequence are 53, 73, 83, 89, 149, 173, 229, 241, 337, 479, 571, 613, 661, 757, 829, 877, 911, 977, 991, ... - M. F. Hasler, Jul 28 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pi Digits.
OEIS index to sequences related to Pi.

Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A053746 (= a(n) + 1: the same with different offset).
Cf. A037000, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A037008 (similar for digits 1, ..., 9 and 0).
Cf. A035117 (first occurrence of at least n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).
Cf. A096755 (first occurrence of exactly n '1's), A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763 (exactly n '9's), A050279 (exactly n '0's).
Cf. A121280 = A068987 - 1: position of "123...n" in Pi's decimals.
Cf. A176341: first occurrence of n in Pi's digits.

Flatten @ Position[ RealDigits[Pi - 3, 10, 500][[1]], 2] (* Robert G. Wilson v, Mar 07 2011 *)

A037001_upto(N=999, d=2)={localprec(N+20); [i-1|i<-[1..#N=digits(Pi\10^-N)], N[i]==d]} \\ M. F. Hasler, Jul 28 2024

a(n) ~ 10*n if Pi is normal, as generally assumed. - M. F. Hasler, Jul 28 2024

A059742 Decimal expansion of e + Pi.

Original entry on oeis.org

5, 8, 5, 9, 8, 7, 4, 4, 8, 2, 0, 4, 8, 8, 3, 8, 4, 7, 3, 8, 2, 2, 9, 3, 0, 8, 5, 4, 6, 3, 2, 1, 6, 5, 3, 8, 1, 9, 5, 4, 4, 1, 6, 4, 9, 3, 0, 7, 5, 0, 6, 5, 3, 9, 5, 9, 4, 1, 9, 1, 2, 2, 2, 0, 0, 3, 1, 8, 9, 3, 0, 3, 6, 6, 3, 9, 7, 5, 6, 5, 9, 3, 1, 9, 9, 4, 1, 7, 0, 0, 3, 8, 6, 7, 2, 8, 3, 4, 9, 5, 4, 0, 9, 6, 1

Offset: 1

Fabian Rothelius, Feb 10 2001

It is not presently known if this number is rational or irrational.

			5.859874482048838473822930854632165381954416493075065395941912220031893...

Harry J. Smith, Table of n, a(n) for n = 1..20000

Cf. A001113, A000796, A058651 (continued fraction).

Digits := 200: with(numtheory): it := evalf((Pi+exp(1))/10, 200): for i from 1 to 20 0 do printf(`%d,`,floor(10*it)): it := 10*it-floor(10*it): od:

RealDigits[E + Pi, 10, 105][[1]] (* Robert G. Wilson v, Sep 24 2004 *)

{ default(realprecision, 20080); x=Pi+exp(1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b059742.txt", n, " ", d)); } \\ Harry J. Smith, May 31 2009

More terms from James Sellers, Feb 13 2001

A090771 Numbers that are congruent to {1, 9} mod 10.

Original entry on oeis.org

1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69, 71, 79, 81, 89, 91, 99, 101, 109, 111, 119, 121, 129, 131, 139, 141, 149, 151, 159, 161, 169, 171, 179, 181, 189, 191, 199, 201, 209, 211, 219, 221, 229, 231, 239, 241, 249, 251, 259, 261, 269, 271, 279, 281

Offset: 1

Giovanni Teofilatto, Feb 07 2004

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2-1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 10). - Bruno Berselli, Nov 17 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A000796, A090298, A005408, A047209, A007310, A019970, A047336, A047522, A091998, A094888, A113801, A175886, A175887, A188593.
Cf. A056020 (n = 1 or 8 mod 9), A175885 (n = 1 or 10 mod 11).
Cf. A045468 (primes), A195142 (partial sums).

a090771 n = a090771_list !! (n-1)
a090771_list = 1 : 9 : map (+ 10) a090771_list
-- Reinhard Zumkeller, Jan 07 2012

Flatten[Table[10n - {9, 1}, {n, 30}]] (* Alonso del Arte, Sep 02 2014 *)
LinearRecurrence[{1,1,-1},{1,9,11},60] (* Harvey P. Dale, Jul 05 2020 *)

a(n)=n\2*10-(-1)^n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = sqrt(40*A057569(n) + 1). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Sep 16 2010 - Nov 17 2010: (Start)
  G.f.: x*(1 + 8*x + x^2)/((1 + x)*(1 - x)^2).
  a(n) = (10*n + 3*(-1)^n - 5)/2.
  a(n) = -a(-n + 1) = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 10.
  a(n) = 10*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
a(n) = 10*n - a(n-1) - 10 (with a(1) = 1). - Vincenzo Librandi, Nov 16 2010
a(n) = sqrt(10*A132356(n-1) + 1). - Ivan N. Ianakiev, Nov 09 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/10)*cot(Pi/10) = A000796 * A019970 / 10 = sqrt(5 + 2*sqrt(5))*Pi/10. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((10*x - 5)*exp(x) + 3*exp(-x))/2. - David Lovler, Sep 03 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(phi+2) (A188593).
Product_{n>=2} (1 + (-1)^n/a(n)) = Pi*phi/5 = A094888/10. (End)

Edited and extended by Ray Chandler, Feb 10 2004

A004603 Expansion of Pi in base 4.

Original entry on oeis.org

3, 0, 2, 1, 0, 0, 3, 3, 3, 1, 2, 2, 2, 2, 0, 2, 0, 2, 0, 1, 1, 2, 2, 0, 3, 0, 0, 2, 0, 3, 1, 0, 3, 0, 1, 0, 3, 0, 1, 2, 1, 2, 0, 2, 2, 0, 2, 3, 2, 0, 0, 0, 3, 1, 3, 0, 0, 1, 3, 0, 3, 1, 0, 1, 0, 2, 2, 1, 0, 0, 0, 2, 1, 0, 3, 2, 0, 0, 2, 0, 2, 0, 2, 2, 1, 2, 1, 3, 3, 0, 3, 0, 1, 3, 1, 0, 0, 0, 0, 2, 0, 0, 2, 3, 2

Offset: 1

N. J. A. Sloane

Theoretically, this sequence could be used to encode a given number of digits of Pi as a DNA sequence, which could then be read back from one helix. The value read back from the other helix would of course depend on the assignment of G, A, C, T to the digits 0, 1, 2, 3. - Alonso del Arte, Nov 07 2011

			3.02100333122220202011220300203103010301...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Francisco Javier Aragón Artacho, 100 billion step walk on the digits of pi
F. J. Aragon Artacho, D. H. Bailey, J. M. Borwein, P. B. Borwein, J. Fountain, and M. Skerritt Walking on numbers: a multiple media mathematics project, 2012.
Elias Bröms, Pictures of Pi

Pi in base b: A004601 (b=2), A004602 (b=3), this sequence (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).
Cf. A007514.
Cf. A004595, A004541. - Jason Kimberley, Dec 01 2012

RealDigits[Pi, 4, 100][[1]]
Table[ResourceFunction["NthDigit"][Pi, n, 4], {n, 1, 100}] (* Joan Ludevid, Jul 04 2022; easy to compute a(10000000)=2 with this function; requires Mathematica 12.0+ *)

a(n) = 2*A004601(2n) + A004601(2n+1). - Jason Kimberley, Nov 08 2012

A037007 Positions of the digit '9' in the decimal expansion of Pi, where positions 0, 1, 2,... correspond to digits 3, 1, 4, ....

Original entry on oeis.org

5, 12, 14, 30, 38, 42, 44, 45, 55, 58, 62, 79, 80, 100, 122, 129, 144, 169, 180, 187, 190, 193, 199, 208, 214, 247, 249, 259, 284, 294, 328, 331, 336, 341, 353, 356, 388, 391, 399, 414, 416, 418, 422, 433, 440, 459, 460, 465, 482, 487, 496, 498, 501, 527

Offset: 1

Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br)

Primes in this sequence are 5, 79, 193, 199, 331, 353, 433, 487, 941, ... - M. F. Hasler, Jul 29 2024

			The first digit '9' occurs in 3.1415926... at the 5th place after the decimal point, whence a(1) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..369 from M. F. Hasler)
Eric Weisstein's World of Mathematics, Pi Digits.
Index entries for sequences related to the number Pi

Cf. A000796 (decimals of Pi).
Cf. A053753 (variant with all values increased by 1).
Cf. A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037008 (similar for digits 1, ..., 8 and 0).
Cf. A048940, A096763 (starting position of at least/exactly n '9's).

Flatten @ Position[ RealDigits[Pi - 3, 10, 500][[1]], 9] (* Robert G. Wilson v, Mar 07 2011 *)

A037007_upto(N=999, d=9)={localprec(N+20); [i-1|i<-[1..#N=digits(Pi\10^-N)], N[i]==d]} \\ M. F. Hasler, Jul 29 2024

a(n) = A053753(n) - 1. - M. F. Hasler, Mar 20 2017

a(n) ~ 10*n if Pi is normal (as generally assumed, but yet unproved). - M. F. Hasler, Jul 29 2024

A060707 Base-60 (Babylonian or sexagesimal) expansion of Pi.

Original entry on oeis.org

3, 8, 29, 44, 0, 47, 25, 53, 7, 24, 57, 36, 17, 43, 4, 29, 7, 10, 3, 41, 17, 52, 36, 12, 14, 36, 44, 51, 50, 15, 33, 7, 23, 59, 9, 13, 48, 22, 12, 21, 45, 22, 56, 47, 39, 44, 28, 37, 58, 23, 21, 11, 56, 33, 22, 40, 42, 31, 6, 6, 3, 46, 16, 52, 2, 48, 33, 24, 38, 33, 22, 1, 0, 1

Offset: 1

Robert G. Wilson v, Feb 05 2001

Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.
Mohammad K. Azarian, The Introduction of Al-Risala al-Muhitiyya: An English Translation, International Journal of Pure and Applied Mathematics, Vol. 57, No. 6, 2009, pp. 903-914.
Mohammad K. Azarian, Al-Kashi's Fundamental Theorem, International Journal of Pure and Applied Mathematics, Vol. 14, No. 4, 2004, pp. 499-509. Mathematical Reviews, MR2005b:01021 (01A30), February 2005, p. 919. Zentralblatt MATH, Zbl 1059.01005.
Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.
Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.

Harry J. Smith, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pi Digits

Cf. A000796, A091649, A159991.

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), this sequence (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60). - Jason Kimberley, Dec 06 2012

Mathematica
```
RealDigits[ Pi, 60, 75][[1]]
```

{ default(realprecision, 17900); x=Pi; for (n=1, 10000, d=floor(x); x=(x-d)*60; write("b060707.txt", n, " ", d)); } \\ Harry J. Smith, Jul 09 2009

A068440 Expansion of Pi in base 15.

Original entry on oeis.org

3, 2, 1, 12, 13, 1, 13, 12, 4, 6, 12, 2, 11, 7, 14, 5, 0, 8, 4, 8, 4, 7, 7, 3, 14, 0, 6, 9, 1, 9, 13, 1, 14, 5, 0, 9, 6, 3, 13, 11, 7, 9, 12, 6, 9, 7, 3, 9, 14, 10, 3, 7, 3, 1, 14, 7, 9, 12, 13, 14, 1, 0, 10, 8, 14, 13, 4, 12, 6, 3, 0, 10, 8, 3, 11, 9, 11, 5, 13, 10, 4, 6, 4, 10, 9, 1, 5, 2

Offset: 1

Benoit Cloitre, Mar 09 2002

			3.21cd1dc46c2b...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), this sequence (b=15), A062964 (b=16), A060707 (b=60).

Cf. A007514.

RealDigits[ N[ Pi, 115], 15] [[1]]
RealDigits[Pi,15,120][[1]] (* Harvey P. Dale, Aug 05 2025 *)

A073233 Decimal expansion of Pi^Pi.

Original entry on oeis.org

3, 6, 4, 6, 2, 1, 5, 9, 6, 0, 7, 2, 0, 7, 9, 1, 1, 7, 7, 0, 9, 9, 0, 8, 2, 6, 0, 2, 2, 6, 9, 2, 1, 2, 3, 6, 6, 6, 3, 6, 5, 5, 0, 8, 4, 0, 2, 2, 2, 8, 8, 1, 8, 7, 3, 8, 7, 0, 9, 3, 3, 5, 9, 2, 2, 9, 3, 4, 0, 7, 4, 3, 6, 8, 8, 8, 1, 6, 9, 9, 9, 0, 4, 6, 2, 0, 0, 7, 9, 8, 7, 5, 7, 0, 6, 7, 7, 4, 8, 5, 4, 3, 6, 8, 1

Offset: 2

Rick L. Shepherd, Jul 21 2002

A weak form of Schanuel's Conjecture implies that Pi^Pi is transcendental--see Marques and Sondow (2012).

			36.4621596072079117709908260226...

Harry J. Smith, Table of n, a(n) for n = 2..20000
D. Marques and J. Sondow, The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture, arXiv:1212.6931 [math.NT], 2012-2013.

Cf. A000796 (Pi), A073234 (Pi^Pi^Pi), A073237 (ceil(Pi^Pi^...^Pi), n Pi's), A073238 (Pi^(1/Pi)), A073239 ((1/Pi)^Pi), A073240 ((1/Pi)^(1/Pi)), A073243 (limit of (1/Pi)^(1/Pi)^...^(1/Pi)), A073236 (Pi analog of A004002).
Cf. A073226 (e^e).
Cf. A049006 (i^i), A116186 (real part of i^i^i).
Cf. A194555 (real part of i^e^Pi).

RealDigits[N[Pi^Pi,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

PARI
```
Pi^Pi
```

{ default(realprecision, 20080); x=Pi^Pi/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b073233.txt", n, " ", d)); } \\ Harry J. Smith, Apr 30 2009

cp's OEIS Frontend

A089491 Decimal expansion of Buffon's constant 3/Pi.

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A004608 Expansion of Pi in base 9.

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A037001 Positions of the digit '2' in the decimal expansion of Pi (where positions 0, 1, 2,... refer to the digits 3, 1, 4,...).

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A059742 Decimal expansion of e + Pi.

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Maple

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A090771 Numbers that are congruent to {1, 9} mod 10.

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Haskell

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A004603 Expansion of Pi in base 4.

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A037007 Positions of the digit '9' in the decimal expansion of Pi, where positions 0, 1, 2,... correspond to digits 3, 1, 4, ....

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