A000796 -id:A000796 - cp's OEIS frontend

A068438 Expansion of Pi in base 13.

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

Offset: 1

Benoit Cloitre, Mar 09 2002

			3.1ac1049052a2c77369c0aa89cc988327829835...

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), this sequence (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).

Cf. A007514.

RealDigits[Pi, 13, 111][[1]] (* slightly modified by Robert G. Wilson v, Dec 13 2017 *)
Table[ResourceFunction["NthDigit"][Pi, n, 13], {n, 1, 111}] (* Joan Ludevid, Oct 11 2022; easy to compute a(10000000)=1 with this function; requires Mathematica 12.0+ *)

A068439 Expansion of Pi in base 14.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Mar 09 2002

			3.1da75cda81375427a40abcb1bd47549c89bcb6...

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), this sequence (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).

Mathematica
```
RealDigits[Pi, 14, 115][[1]]
```

A096761 Position of first occurrence of exactly n consecutive sevens in a row in the decimal expansion of Pi.

Original entry on oeis.org

13, 559, 4575, 1589, 162248, 399579, 3346228, 82144203, 24658601, 22869046249, 165431035708, 368299898266, 10541103245815, 14793486898235, 46970519777308

Offset: 1

Robert G. Wilson v, Jul 07 2004

Differs from A050286 from a(3) > a(4) on. - M. F. Hasler, Mar 18 2017
a(11) > 99*10^9. - Giovanni Resta, Oct 02 2019
a(15) > 22*10^12. - Dmitry Petukhov, Jan 27 2020
a(16) > 50*10^12. - Dmitry Petukhov, Oct 30 2021

David G. Andersen, The Pi-Search Page.
Timothy Mullican, 50 trillion digits of pi
Peter Trüb, 22.4 trillion 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. A000796 (decimal expansion (or digits) of Pi).

Edited by M. F. Hasler, Mar 19 2017
a(10) from Giovanni Resta, Oct 02 2019
a(11)-a(13) added by Dmitry Petukhov, Jan 13 2020
a(14) from Dmitry Petukhov, Jan 27 2020
a(15) from Dmitry Petukhov, Oct 30 2021

A096762 Position of first occurrence of exactly n consecutive '8's in a row in the decimal expansion of Pi.

Original entry on oeis.org

11, 34, 4985, 4751, 213245, 222299, 4722613, 239798471, 46663520, 3040319543, 159999448572, 1141385905180, 2164164669332, 91250566353705

Offset: 1

Robert G. Wilson v, Jul 07 2004

a(8) > 2*10^8, a(9) = 46663520, a(10) = 3040319543.

Differs from A050287 from a(3) > A050287(3) = A050287(4) = a(4) on. - M. F. Hasler, Mar 19 2017

David G. Andersen, The Pi-Search Page.
Peter Trüb, 22.4 trillion digits of pi

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), A050279 (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.

Edited by M. F. Hasler, Mar 19 2017
a(8) via SubIdiom.com/pi search engine from M. F. Hasler, Apr 13 2019
a(11)-a(13) added by Dmitry Petukhov, Dec 30 2019
a(14) from Dmitry Petukhov, Sep 20 2022

A176341 a(n) = the location of the first appearance of the decimal expansion of n in the decimal expansion of Pi.

Original entry on oeis.org

32, 1, 6, 0, 2, 4, 7, 13, 11, 5, 49, 94, 148, 110, 1, 3, 40, 95, 424, 37, 53, 93, 135, 16, 292, 89, 6, 28, 33, 186, 64, 0, 15, 24, 86, 9, 285, 46, 17, 43, 70, 2, 92, 23, 59, 60, 19, 119, 87, 57, 31, 48, 172, 8, 191, 130, 210, 404, 10, 4, 127, 219, 20, 312, 22, 7, 117, 98, 605, 41

Offset: 0

Daniel E. Loeb, Apr 15 2010

It is unknown whether Pi is a normal number. If it is (at least in base 10) then this sequence is well defined.

The numbers a(n) refer to the position of the initial digit of n in the decimal expansion of Pi, where "3" is at position a(3)=0, "1" is at position a(1)=1, etc. This is also the numbering scheme used on the "Pi search page" cited among the LINKS. See A232013 for a sequence based on iterations of this one. See A032445 for a variant of the present sequence, where numbering starts at one. - M. F. Hasler, Nov 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dave Andersen, Search within first 200,000,000 digits of pi
Michael D. Huberty, Ko Hayashi & Chia Vang, First 100,000 digits of pi
Simon Plouffe, First 50,000,000 digits of pi

Cf. A000796, A011545, A014777, A032445, A232013.

p=ToString[FromDigits[RealDigits[N[Pi, 10^4]][[1]]]]; Do[Print[StringPosition[p, ToString[n]][[1]][[1]] - 1], {n, 0, 100}] (* Vincenzo Librandi, Apr 17 2017 *)
With[{pid=RealDigits[Pi,10,800][[1]]},Flatten[Table[ SequencePosition[ pid,IntegerDigits[n],1],{n,0,70}],1]][[All,1]]-1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 27 2019 *)

A176341(n)=my(L=#Str(n));n=Mod(n,10^L);for(k=L-1,9e9,Pi\.1^k-n||return(k+1-L)) \\ Make sure to use sufficient realprecision, e.g. via \p999. - M. F. Hasler, Nov 16 2013

pi = "314159265358979323846264338327950288419716939937510582097494459230..."
[ pi.find(str(i)) for i in range(10000) ]

a(n) = A032445(n)-1. - M. F. Hasler, Nov 16 2013

a(n) = 0 if n is in A011545, otherwise a(n) = A014777(n). - Pontus von Brömssen, Aug 31 2024

A011546 Decimal expansion of Pi rounded to n places.

Original entry on oeis.org

3, 31, 314, 3142, 31416, 314159, 3141593, 31415927, 314159265, 3141592654, 31415926536, 314159265359, 3141592653590, 31415926535898, 314159265358979, 3141592653589793, 31415926535897932, 314159265358979324, 3141592653589793238, 31415926535897932385, 314159265358979323846

Offset: 0

N. J. A. Sloane

Scherzer (2012) writes: "The 17th-most common 10-digit password is 3141592654 (for you non-math nerds, those are the first [ten] digits of Pi)." The information comes from an analysis of expired ATM PIN codes conducted by Nick Berry of Data Genetics. - Alonso del Arte, Sep 21 2012

			a(4) = floor(10^4 * Pi + 0.5) = 31416.

Paolo Xausa, Table of n, a(n) for n = 0..995
Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, (The Treatise on the Circumference), Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.
Lisa Scherzer, "Cracking Your PIN Code: Easy as 1-2-3-4", The Exchange, Sep 21 2012.

Cf. A000796, A011552, A011544, A011548.

a:= proc(n) Digits:= n+20;
       round(10^n * Pi)
    end:
seq(a(n), n=0..20); # Alois P. Heinz, Mar 11 2016

Module[{nn=20,pid},pid=RealDigits[Pi,10,nn+2][[1]];Table[Floor[ (FromDigits[ Take[pid,n+1]])/10+1/2],{n,nn}]] (* Harvey P. Dale, Oct 09 2017 *)
Round[Pi*10^Range[0, 20]] (* Paolo Xausa, Jul 08 2025 *)

a(n)=round(Pi*10^n) \\ Charles R Greathouse IV, Sep 21 2012

a(n) = floor(10^n * Pi + 0.5).

A046974 Partial sums of digits of decimal expansion of Pi.

Original entry on oeis.org

3, 4, 8, 9, 14, 23, 25, 31, 36, 39, 44, 52, 61, 68, 77, 80, 82, 85, 93, 97, 103, 105, 111, 115, 118, 121, 129, 132, 134, 141, 150, 155, 155, 157, 165, 173, 177, 178, 187, 194, 195, 201, 210, 213, 222, 231, 234, 241, 246, 247, 247, 252, 260, 262

Offset: 0

N. J. A. Sloane

a(n) = A007953(A011545(n)). - Reinhard Zumkeller, Oct 30 2003
The partial sums to 10^k, k>=0: 4, 44, 480, 4479, 44897, 449336, 4499937, 45002885, 449989731, .... - Robert G. Wilson v, Sep 16 2007
If the sequence were to start with an initial term a(0) = 0, its first differences would reproduce the complete sequence of digits of Pi. - M. F. Hasler, Jan 19 2015

Mital Ashok, Table of n, a(n) for n = 0..999

Cf. A000796, A089290.

Rest@ FoldList[ Plus, 0, First@ RealDigits[Pi, 10, 58]] (* Robert G. Wilson v, Sep 16 2007 *)
Accumulate[RealDigits[Pi,10,60][[1]]] (* Harvey P. Dale, Mar 11 2013 *)

A092731 Decimal expansion of Pi^5.

Original entry on oeis.org

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

Offset: 3

Mohammad K. Azarian, Apr 12 2004

			306.0196847852814532

G. C. Greubel, Table of n, a(n) for n = 3..10000
Jean-Christophe Pain, The fifth power of Pi: new series representation involving the golden ratio and an application in physics, 2022.
Jean-Christophe Pain, New series representations for any positive power of Pi from a relation involving trigonometric functions, 2208.02624 [math.NT], 2022.
Kh. Hessami Pilehrood and Tatiana Hessami Pilehrood, Series acceleration formulas for beta values, Discr. Math. Theor. Comp. Sci. 12 (2) (2010) 223-236.
Index entries for transcendental numbers

Cf. A000796, A002161, A019692, A091925, A092735, A000364, A002437.

R:= RealField(100); (Pi(R))^5; // G. C. Greubel, Mar 09 2018

RealDigits[Pi^5, 10, 100][[1]] (* G. C. Greubel, Mar 09 2018 *)

PARI
```
Pi^5 \\ G. C. Greubel, Mar 09 2018
```

From Peter Bala, Oct 31 2019: (Start)
Pi^5 = (4!/(2*305)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/6)^5 + 1/(n + 5/6)^5 ), where 305 = ((3^5 + 1)/4)*A000364(2) = A002437(2).
Pi^5 = (4!/(2*3905)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/10)^5 - 1/(n + 3/10)^5 - 1/(n + 7/10)^5 + 1/(n + 9/10)^5 ), where 3905 = ((5^5 - 1)/4)*A000364(2).
Cf. A019692, A091925 and A092735. (End)

A057679 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where 3 is the first digit.

Original entry on oeis.org

5, 242424, 271070, 9292071, 29133316, 70421305, 215817165252, 649661007154

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)
a(2) is not the first occurrence of 242424 in Pi (which is at position 242422) but the second. - Hans Havermann, Jul 26 2014
a(9) is greater than 5 * 10^13. - Kang Seonghoon, Nov 02 2020

			5 is a term because 5 is the 5th digit of Pi (3.1415...).

Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (2020).
Google, 50 trillion digits of pi (2020).

Cf. A000796, A057680, A064810, A109514.

StringsinPi[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[Pi,10,cc] // First, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] (* For example, StringsinPi[6] returns all 6-digit members of the sequence. - Colin Rose, Mar 15 2006 *)
dpi = RealDigits[Pi, 10, 10000010][[1]]; Select[Range[10000000], FromDigits[Take[dpi, {#, # - 1 + IntegerLength[#]}]] == # &] (* Vaclav Kotesovec, Feb 18 2020 *)

a(4)-a(6) from Colin Rose, Mar 15 2006
a(7) from Alan Eliasen, May 10 2013
a(8) from Alan Eliasen, Jun 06 2013
Name clarified by Kang Seonghoon, Nov 02 2020

A014493 Odd triangular numbers.

Original entry on oeis.org

1, 3, 15, 21, 45, 55, 91, 105, 153, 171, 231, 253, 325, 351, 435, 465, 561, 595, 703, 741, 861, 903, 1035, 1081, 1225, 1275, 1431, 1485, 1653, 1711, 1891, 1953, 2145, 2211, 2415, 2485, 2701, 2775, 3003, 3081, 3321, 3403, 3655, 3741, 4005, 4095, 4371, 4465, 4753, 4851

Offset: 1

Mohammad K. Azarian

Odd numbers of the form n*(n+1)/2.
For n such that n(n+1)/2 is odd see A042963 (congruent to 1 or 2 mod 4).
Even central polygonal numbers minus 1. - Omar E. Pol, Aug 17 2011
Odd generalized hexagonal numbers. - Omar E. Pol, Sep 24 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 68.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
D. H. Lehmer, Recurrence formulas for certain divisor functions, Bull. Amer. Math. Soc., Vol. 49, No. 2 (1943), pp. 150-156.
Eric Weisstein's World of Mathematics, Triangular Number.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A000796, A014494, A019669, A042963, A067589, A128880.

List([1..50], n -> (2*n-1)*(2*n-1-(-1)^n)/2); # G. C. Greubel, Feb 09 2019

[(2*n-1)*(2*n-1-(-1)^n)/2: n in [1..50]]; // Vincenzo Librandi, Aug 18 2011

[(2*n-1)*(2*n-1-(-1)^n)/2$n=1..50]; # Muniru A Asiru, Mar 10 2019

Select[ Table[n(n + 1)/2, {n, 93}], OddQ[ # ] &] (* Robert G. Wilson v, Nov 05 2004 *)
LinearRecurrence[{1,2,-2,-1,1},{1,3,15,21,45},50] (* Harvey P. Dale, Jun 19 2011 *)

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

def A014493(n): return ((n<<1)-1)*(n-(n&1^1)) # Chai Wah Wu, Feb 12 2023

[(2*n-1)*(2*n-1-(-1)^n)/2 for n in (1..50)] # G. C. Greubel, Feb 09 2019

From Ant King, Nov 17 2010: (Start)
a(n) = (2*n-1)*(2*n - 1 - (-1)^n)/2.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)
G.f.: x*(1 + 2*x + 10*x^2 + 2*x^3 + x^4)/((1+x)^2*(1-x)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = A000217(A042963(n)). - Reinhard Zumkeller, Feb 14 2012, Oct 04 2004
a(n) = A193868(n) - 1. - Omar E. Pol, Aug 17 2011
Let S = Sum_{n>=0} x^n/a(n), then S = Q(0) where Q(k) = 1 + x*(4*k+1)/(4*k + 3 - x*(2*k+1)*(4*k+3)^2/(x*(2*k+1)*(4*k+3) + (4*k+5)*(2*k+3)/Q(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 27 2013
E.g.f.: (2*x^2+x+1)*cosh(x)+x*(2*x-1)*sinh(x)-1. - Ilya Gutkovskiy, Apr 24 2016
Sum_{n>=1} 1/a(n) = Pi/2 (A019669). - Robert Bilinski, Jan 20 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2). - Amiram Eldar, Mar 06 2022

More terms from Erich Friedman

cp's OEIS Frontend

A068438 Expansion of Pi in base 13.

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Mathematica

A068439 Expansion of Pi in base 14.

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Mathematica

A096761 Position of first occurrence of exactly n consecutive sevens in a row in the decimal expansion of Pi.

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A096762 Position of first occurrence of exactly n consecutive '8's in a row in the decimal expansion of Pi.

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A176341 a(n) = the location of the first appearance of the decimal expansion of n in the decimal expansion of Pi.

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Mathematica

PARI

Python

Formula

A011546 Decimal expansion of Pi rounded to n places.

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Maple

Mathematica

PARI

Formula

A046974 Partial sums of digits of decimal expansion of Pi.

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Mathematica

A092731 Decimal expansion of Pi^5.

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Magma

Mathematica

PARI

Formula

A057679 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where 3 is the first digit.