A000796 -id:A000796 - cp's OEIS frontend

A004604 Expansion of Pi in base 5.

3, 0, 3, 2, 3, 2, 2, 1, 4, 3, 0, 3, 3, 4, 3, 2, 4, 1, 1, 2, 4, 1, 2, 2, 4, 0, 4, 1, 4, 0, 2, 3, 1, 4, 2, 1, 1, 1, 4, 3, 0, 2, 0, 3, 1, 0, 0, 2, 2, 0, 0, 3, 4, 4, 4, 1, 3, 2, 2, 1, 1, 0, 1, 0, 4, 0, 3, 3, 2, 1, 3, 4, 4, 0, 0, 4, 3, 2, 4, 4, 4, 0, 1, 4, 4, 1, 0, 4, 2, 3, 3, 4, 1, 3, 3, 0, 1, 1, 3, 2

Offset: 1

N. J. A. Sloane

			3.03232214303343241124122404140231421114...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wadim Zudilin, A BBP-style computation for π in base 5, arXiv:2409.10097 [math.NT], 2024.

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).

Cf. A007514.

RealDigits[Pi, 5, 100][[1]]
Table[ResourceFunction["NthDigit"][Pi, n, 5], {n, 1, 100}] (* Joan Ludevid, Aug 17 2022;easy to compute a(10000000)=0 with this function;requires Mathematica 12.0+ *)

A004606 Expansion of Pi in base 7.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.06636514320361341102634022446522266435...

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), this sequence (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.

RealDigits[Pi, 7, 105][[1]]
Table[ResourceFunction["NthDigit"][Pi, n, 7], {n, 1, 105}] (* Joan Ludevid, Sep 13 2022; easy to compute a(10000000)=5 with this function;requires Mathematica 12.0+ *)

A006941 Expansion of Pi in base 8.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.1103755242102643021514230630505600670...

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 614.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to the number Pi

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), this sequence (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.

convert(evalf(Pi), octal, 120);  # Alois P. Heinz, Dec 16 2018

RealDigits[ N[ Pi, 105], 8] [[1]]
Table[ResourceFunction["NthDigit"][Pi, n, 8], {n, 1, 105}] (* Joan Ludevid, Sep 13 2022; easy to compute a(10000000)=1 with this function; requires Mathematica 12.0+ *)

a(n) = 4*A004601(3n) + 2*A004601(3n+1) + 1*A004601(3n+2). - Jason Kimberley, Nov 06 2012

More terms from Michel ten Voorde, Apr 14 2001

A091476 Decimal expansion of Pi^2/4.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 13 2004

			2.46740110027233965470862274996903778...

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Ben Hambrecht and Grant Sanderson, The stunning geometry behind this surprising equation, 3Blue1Brown video (2018).
Josef Hofbauer, A simple proof of 1 + 1/2^2 + 1/3^2 + ... = Pi^2/6 and related identities, The American Mathematical Monthly 109:2 (2002), pp. 196-200.
Michael Penn, Neat ways to solve complicated limits., YouTube video, 2023.
T. Piezas III, Golden ratio and nested radicals
Johan Wästlund, Summing inverse squares by euclidean geometry
Eric Weisstein's World of Mathematics, Definite Integral
H. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theoretical Computer Science 3(4) (1999), 189-192.
Index entries for transcendental numbers

Cf. A002388, A013661, A019669.

evalf(Pi^2/4, 120); # Muniru A Asiru, Sep 18 2018

First[RealDigits[Pi^2/4,10,100]] (* Paolo Xausa, Oct 31 2023 *)

Pi^2/4 \\ Charles R Greathouse IV, Mar 02 2018

2*sumpos(n=1,(2*n-1)^-2) \\ Charles R Greathouse IV, Mar 02 2018

Equals Integral_{x=0..Pi} x*sin(x)/(1+cos(x)^2) dx.
Equals Integral_{x=0..1} log((1+x)/(1-x))/x dx. - Jean-François Alcover, May 13 2013
Equals Integral_{x=0..oo} K_0(x)^2 dx, where K_0 is a modified Bessel function (see Gradstein-Ryshik 6.576.4). - R. J. Mathar, Oct 09 2015
Equals A003881 * A000796. - R. J. Mathar, Oct 09 2015
Equals ... + (-5)^-2 + (-3)^-2 + (-1)^-2 + 1^-2 + 3^-2 + 5^-2 + .... - Charles R Greathouse IV, Mar 02 2018
From A.H.M. Smeets, Sep 18 2018: (Start)
Equals A102753/2.
Equals 2*Sum_{k > 0} 1/(2*k - 1)^2. (End)
Pi^2/4 = Integral_{x = 0..oo} x/sinh(x) dx. More generally, Pi^2/4 = 2*(1 + 1/3^2 + ... + 1/(2*n-1)^2) + Integral_{x = 0..oo} exp(-2*n*x)*x/sinh(x). - Peter Bala, Nov 05 2019
Equals Integral_{x=0..oo} log(x)/(x^2 - 1) dx. - Amiram Eldar, Aug 12 2020
Equals Sum_{n >= 0} 2^(n+1)/((n+1)^2*binomial(2*n+1,n)). See my entry in A002544 dated Apr 18 2017. Cf. A253191. - Peter Bala, Jan 30 2023
From Peter Bala, Nov 16 2023: (Start)
Pi^2/4 = 16*Sum_{k >= 1} k^2/(4*k^2 - 1)^2 = (2*16^2)*Sum_{k >= 1} k^2/((4*k^2 - 1)*(4*k^2 - 9))^2.
The general result, which can be proved using the WZ method (see Wilf for examples of this method), is that for n >= 0 there holds
Pi^2/4 = 16^(n+1)*(2*n + 1)*(2*n)!^4/(4*n)! * Sum_{k >= 1} k^2/( (4*k^2 - 1)*(4*k^2 - 9)*...*(4*k^2 - (2*n+1)^2) )^2. (End)
Equals Re(Polylog(2, 2)). - Mohammed Yaseen, Jul 03 2024
From A.H.M. Smeets, Apr 10 2025: (Start)
Let X(p,q) be the p-th smallest zero of the Laguerre polynomial of order q.
Equals lim_{k -> oo} X(k,k^2).
Equals lim_{q -> oo} X(1,q)*q.
Equals lim_{k -> oo} X(k,k^4)*sqrt(k).
Equals lim_{k -> oo} X(k^3,k^4)/sqrt(k).
More general, let P = log_q(p^2/q), then, for any p, 0 < p <= q, equals lim_{q -> oo} X(p,q)/q^P. (End)
Equals Integral_{x=-1..1} -log(abs(x))/(1 - x^2) dx. - Kritsada Moomuang, May 28 2025

A102753 Decimal expansion of (Pi^2)/2.

Original entry on oeis.org

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

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 10 2005

Also equals the area under the peak-shaped even function f(x)=x/sinh(x).
Proof: For the upper half of the integral, write f(x) = 2x*exp(-x)/(1-exp(-2x)) = sum_{k=1..infinity} 2x*exp(-(2k-1)x) and integrate term by term from zero to infinity. - Stanislav Sykora, Nov 01 2013
Volume of the 4-dimensional unit sphere; the volume of the n-dimensional unit sphere is Pi^(n/2)/gamma(n/2+1) (see n-ball link and A164103). - Rick L. Shepherd, Jun 22 2017
Pi^2/2 is the squared side-length of a square with diagonal Pi. - Wesley Ivan Hurt, Jan 28 2022

			4.9348022005446793094172454999380755676568497036203953132066746881100\ 224112096026215008867018592761159120129568870115720388....

J. Rivaud, Analyse, Séries, Equations différentielles, Mathématiques Supérieures et Spéciales, Premier Cycle Universitaire, Vuibert, 1981, Exercice 2, p. 135.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, 1986, p. 53.

G. C. Greubel, Table of n, a(n) for n = 1..10000
T. Amdeberhan, L. Medina and V. H. Moll, The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals, equation 2.39, arXiv:0705.2379 [math.CA], 2007.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27
Eric Weisstein's World of Mathematics, Hypersphere.
Eric Weisstein's World of Mathematics, Trigamma Function.
Wikipedia, Hypersphere.
Wikipedia, Volume of an n-ball.
Index entries for transcendental numbers

Cf. A002388, A000796, A019699, A026424, A164103, A164105, A164106, A164108, A248359, A276023.

RealDigits[Pi^2/2, 10, 111][[1]] (* Robert G. Wilson v, Dec 15 2005 *)

PARI
```
Pi^2/2 \\ Michel Marcus, Sep 04 2015
```

Equals psi_1(1/2), where psi_1(x) is the second logarithmic derivative of GAMMA(x).
Equals the volume of revolution of the sine or cosine curve for one half period, Integral_{0,Pi} Sin(x)^2 dx. - Robert G. Wilson v, Dec 15 2005
Equals Sum_{k >=1} 4^k/(k^2*binomial(2*k,k)) [Amdeberhan, Sprugnoli]. - R. J. Mathar, Sep 28 2007
Equals 4*Sum_{k >=1} 1/(2k-1)^2 [Wells].
From  Peter Bala, Nov 05 2019: (Start)
Pi^2/2 = Integral_{x = 0..inf} cosh(x)*x^2/sinh(x)^2 dx.
Pi^2/2 = 5*sum_{k >= 0} binomial(2*k,k)(-1/16)^k*1/(2*k+1)^2.
Pi^2/2 = 10*Integral_{x = 0..1/2}  1/x*log(x + sqrt(1 + x^2)) dx. (End)
Pi^2/20 = 0.1 * Pi^2/2 = Sum_{k>=1} 1/A026424(k)^2. - Amiram Eldar, Aug 17 2020
Conjecture: Pi^2/2 = Sum_{n = -oo..oo} ( cos(Pi*sqrt(n^2+1)) - cos(Pi*n) ) (using the Eisenstein summation convention). - Peter Bala, Oct 08 2021
Pi^2/2 = Integral_{x = -oo..oo} x/sinh(x) dx (see Rivaud reference). - Bernard Schott, Jan 28 2022

A032445 Number the digits of the decimal expansion of Pi: 3 is the first, 1 is the second, 4 is the third and so on; a(n) gives the starting position of the first occurrence of n.

Original entry on oeis.org

33, 2, 7, 1, 3, 5, 8, 14, 12, 6, 50, 95, 149, 111, 2, 4, 41, 96, 425, 38, 54, 94, 136, 17, 293, 90, 7, 29, 34, 187, 65, 1, 16, 25, 87, 10, 286, 47, 18, 44, 71, 3, 93, 24, 60, 61, 20, 120, 88, 58, 32, 49, 173, 9, 192, 131, 211, 405, 11, 5, 128, 220, 21, 313, 23, 8, 118, 99, 606

Offset: 0

Jeff Burch, Paul Simon (paulsimn(AT)microtec.net)

See A176341 for a variant counting positions starting with 0, and A232013 for a sequence based on iterations of A176341. - M. F. Hasler, Nov 16 2013

			a(10) = 50 because the first "10" in the decimal expansion of Pi occurs at digits 50 and 51: 31415926535897932384626433832795028841971693993751058209749445923...

T. D. Noe, Table of n, a(n) for n=0..9999
M. J. Halm, More Sequences, Mpossibilities 83, April 2003.
Eric Weisstein's World of Mathematics, Constant Digit Scanning
Eric Weisstein's World of Mathematics, Pi Digits

Cf. A000796 (decimal expansion of Pi).
Cf. A080597 (terms from the decimal expansion of Pi which include every combination of n digits as consecutive subsequences).
Cf. A032510 (last string seen when scanning the decimal expansion of Pi until all n-digit strings have been seen).
Cf. A064467 (primes in Pi).

p = ToString[FromDigits[RealDigits[N[Pi, 10^4]][[1]]]]; Do[Print[StringPosition[p, ToString[n]][[1]][[1]]], {n, 1, 100}]
With[{pi=RealDigits[Pi,10,1000][[1]]},Transpose[Flatten[Table[ SequencePosition[ pi,IntegerDigits[n],1],{n,0,70}],1]][[1]]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Dec 01 2015 *)

A032445(n)=my(L=#Str(n)); n=Mod(n, 10^L); for(k=L-1, 9e9, Pi\.1^k-n||return(k+2-L)) \\ Make sure to use sufficient realprecision, e.g. via \p999. - M. F. Hasler, Nov 16 2013

a(n) = A176341(n)+1. - M. F. Hasler, Nov 16 2013

More terms from Simon Plouffe. Corrected by Michael Esposito and Michelle Vella (michael_esposito(AT)oz.sas.com).

More terms from Robert G. Wilson v, Oct 04 2001

A035117 a(n) is the starting position of the first occurrence of a string of at least n 1's in the decimal expansion of Pi.

Original entry on oeis.org

1, 94, 153, 12700, 32788, 255945, 4657555, 159090113, 812432526, 3961184001, 15647738228, 1041032609981, 3907688331257, 68635742334547

Offset: 1

Leonardo Bitran (lbitran(AT)reuna.cl)

Presently identical to A096755, which is the first occurrences of exactly n 1's in the digits of Pi. Will differ as soon as there's some a(n) = a(n+1) and equivalently, A035117(n) > A035117(n+1). - M. F. Hasler, Mar 17 2017

David G. Andersen, The Pi-Search Page.
PI-world Site, The digits and Statistics for 12 trillion digits of PI [archived page]
Eric Weisstein's World of Mathematics, Pi Digits.
Yasumasa Kanada Laboratory Home Page, Computer Centre, The University of Tokyo, Statistical Distribution Information

Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A035117 (this), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).
Cf. A096755 (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).

More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007
Edited, after re-establishing A096755, by M. F. Hasler, Mar 17 2017
a(11) from Giovanni Resta, Sep 30 2019
a(12) from Yasumasa Kanada Laboratory, 2002 and a(13) from Shigeru Kondo, 2011, added by Dmitry Petukhov, Dec 27 2019
a(14) from Dmitry Petukhov, Sep 19 2022

A048940 Starting position of the first occurrence of a string of at least n '9's in the decimal expansion of Pi.

Original entry on oeis.org

5, 44, 762, 762, 762, 762, 1722776, 36356642, 564665206, 20148132310, 27014073304, 897831316556, 5758910552709, 5758910552709

Offset: 1

Eric W. Weisstein

a(10) > 11*10^9 - 1. - Eric W. Weisstein, Jul 20 2013
a(15) > 22*10^12. - Dmitry Petukhov, Jan 29 2020
Pi digits 3,1,4,... are indexed 0,1,2,... Note that this is different from other sequences such as A049522, A084073 which use indices 1,2,3,... For example, the position of the curious accumulation of six 9s is called 762 here; the same position is called 763 in A049522, A084073. - Jeppe Stig Nielsen, Aug 21 2017

David G. Andersen, The Pi-Search Page. (Yields, as of today, an incorrect result of 66780105 for the first occurrence of eight "9"s. - _M. F. Hasler_, Mar 19 2017)
Yasumasa Kanada Laboratory Home Page, Computer Centre, The University of Tokyo, Statistical Distribution Information
PI-world Site, The digits and Statistics for 12 trillion digits of PI [archived page]
Eric Weisstein's World of Mathematics, Feynman Point
Eric Weisstein's World of Mathematics, Pi Digits

Cf. A000796: Decimal expansion (or digits) of Pi.
First occurrence of n times the same digit: A035117 (n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).
First occurrence of exactly n times the same digit: A096755 (exactly n '1's), A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763 (exactly n '9's), A050279 (exactly n '0's).
First occurrence of n: A176341; of concatenate(1,...,n): A121280 = A068987 - 1.
Cf. A049522, A084073.

Module[{m, nn = 7}, m = First@ RealDigits@ N[Pi, 10^nn]; Array[ SequencePosition[m, ConstantArray[9, #]][[1, 1]] - 1 &, nn]] (* Michael De Vlieger, Mar 20 2017 *)

More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
Edited by M. F. Hasler, Mar 19 2017
a(10)-a(11) from Giovanni Resta, Sep 30 2019
a(12) from Yasumasa Kanada Laboratory, 2002 and a(13)-a(14) from Shigeru Kondo, 2011 added by Dmitry Petukhov, Dec 23 2019

A050279 a(n) is the starting position of the first occurrence of a string of at least n '0's in the decimal expansion of Pi.

Original entry on oeis.org

32, 307, 601, 13390, 17534, 1699927, 3794572, 172330850, 2542542102, 8324296435, 371247087572, 1755524129973, 3186699229890, 6381820482331

Offset: 1

Eric W. Weisstein

At least up to a(10), also the starting position of the first occurrence of a string of exactly n '0's in the decimal expansion of Pi, cf. A096764. - M. F. Hasler, Mar 19 2017, edited Sep 03 2017

a(15) > 22*10^12. - Dmitry Petukhov, Jan 28 2020

Shigeru Kondo, calculation of Pi to 12.8 * 10^9 digits, using the program PiFast of Xavier Gourdon

David G. Andersen, The Pi-Search Page.
Peter Trüb, 22.4 trillion digits of pi
Eric Weisstein's World of Mathematics, Pi Digits

See A096764 for another version.
Cf. A000796: Decimal expansion (or digits) of Pi.
First occurrence of exactly n times the same digit: A096755 (exactly n '1's), A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763 (exactly n '9's), A096764 (exactly n '0's).
First occurrence of n times the same digit: A035117 (n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).
First occurrence of concatenate(1,...,n): A121280 = A068987 - 1.

More terms from Colin B. Martin (martinc(AT)ram.net.au), Nov 25 2001
Edited by N. J. A. Sloane at the suggestion of M. F. Hasler, Aug 24 2007
Edited by M. F. Hasler, Mar 19 2017
Definition modified by N. J. A. Sloane, Sep 03 2017
a(11)-a(14) added by Dmitry Petukhov, Jan 12 2020

A057680 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where the initial digit 3 is at position 0.

Original entry on oeis.org

1, 16470, 44899, 79873884, 711939213, 36541622473, 45677255610, 62644957128, 656430109694

Offset: 1

Mike Keith, Oct 19 2000

The average number of matches of length "n" digits is exactly 0.9.  That is, we expect 0.9 matches with 1 digit, 0.9 matches with 2 digits, etc.  Increasing the number of digits by a factor of 10 means that we expect to find 0.9 new matches.  Increasing the search from 10^11 to 10^12 (which includes 10 times as much work) would thus only expect to find 0.9 new matches. - Alan Eliasen, May 01 2013 (corrected by Michael Beight, Mar 21 2020)
Consequently, with the second Borel-Cantelli lemma, the expected number of terms in this sequence is infinite with probability 1. (Of course the sequence is not random, but almost all of the sequences corresponding to randomly-chosen real numbers in place of Pi have infinitely many terms.) - Charles R Greathouse IV, Apr 29 2013
a(1) & a(5) are the first occurrences in Pi of their respective strings; a(2) & a(4) are the second occurrences; a(3) is the fourth occurrence. - Hans Havermann, Jul 27 2014
A near-miss '043611' occurs at position 43611. - S. Alwin Mao, Feb 18 2020
a(10) > 5 * 10^13. - Kang Seonghoon, Nov 02 2020
Has no terms in common with A037008 (but see Mao comment above). - Charles R Greathouse IV, Jun 21 2022

			1 is a term because the string of digits '1' occurs as the 1st digit after the decimal point.
Similarly, 16470 is a term because the string of digits '16470' occurs starting at position 16470 (after the decimal point) in the digits of Pi (although it already occurs earlier at position 1602). - _M. F. Hasler_, Jul 29 2024

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

David G. Andersen, The Pi-Search Page.
Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (2020).
Google, 50 trillion digits of pi (2020).

Cf. A000796 (decimal digits of Pi), A057679 (variant where position 1 refers to the initial digit 3), A064810 (variant where position 0 refers to the first digit after the decimal point), A109513 (variant using chunks of m digits).

StringsinPiAfterPoint[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[Pi,10,cc] // First // Rest, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] (* For example, StringsinPiAfterPoint[5] returns all 5-digit members of the sequence. - Colin Rose, Mar 15 2006 *)
Do[If[RealDigits[Pi,10,a=i+IntegerLength@i-1,-1][[1,i;;a]]==IntegerDigits@i,Print@i],{i,50000}] (* Giorgos Kalogeropoulos, Feb 21 2020 *)

A057680_row(r=5)={my(M=10^r, R=[]); for(n=M\10, M-1, localprec(n+r); Pi\10^(1-r-n)%M==n && !print1(n",") && R=concat(R,n));R} \\ prints & returns the r-digit terms. - M. F. Hasler, Jul 29 2024

More terms from Colin Rose, Mar 15 2006
a(5) from Nathaniel Johnston, Nov 12 2010
a(6)-a(8) from Alan Eliasen, May 01 2013
a(9) from Alan Eliasen, Jun 06 2013
Name clarified by Kang Seonghoon, Nov 02 2020
Edited by M. F. Hasler, Jul 29 2024

cp's OEIS Frontend

A004604 Expansion of Pi in base 5.

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A004606 Expansion of Pi in base 7.

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A006941 Expansion of Pi in base 8.

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Maple

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A091476 Decimal expansion of Pi^2/4.

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A102753 Decimal expansion of (Pi^2)/2.

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A032445 Number the digits of the decimal expansion of Pi: 3 is the first, 1 is the second, 4 is the third and so on; a(n) gives the starting position of the first occurrence of n.

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A035117 a(n) is the starting position of the first occurrence of a string of at least n 1's in the decimal expansion of Pi.

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A048940 Starting position of the first occurrence of a string of at least n '9's in the decimal expansion of Pi.