A004602 -id:A004602

A004605

Expansion of Pi in base 6.

+10
23

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

EXAMPLE

3.05033005141512410523441405312532110230...

MATHEMATICA

RealDigits[Pi, 6, 105][[1]]

Table[ResourceFunction["NthDigit"][Pi, n, 6], {n, 1, 105}] (* Joan Ludevid , Aug 17 2022; easy to compute a(10000000)=0 with this function; requires Mathematica 12.0+ *)

CROSSREFS

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

KEYWORD

nonn,base,cons,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A004604

Expansion of Pi in base 5.

+10
22

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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.

EXAMPLE

3.03232214303343241124122404140231421114...

MATHEMATICA

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+ *)

CROSSREFS

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.

KEYWORD

nonn,base,cons,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A004606

Expansion of Pi in base 7.

+10
22

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

EXAMPLE

3.06636514320361341102634022446522266435...

MATHEMATICA

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+ *)

CROSSREFS

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.

KEYWORD

nonn,base,cons,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A006941

Expansion of Pi in base 8.
(Formerly M2208)

+10
22

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

REFERENCES

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).

LINKS

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

Index entries for sequences related to the number Pi

FORMULA

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

EXAMPLE

3.1103755242102643021514230630505600670...

MAPLE

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

MATHEMATICA

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+ *)

CROSSREFS

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.

KEYWORD

nonn,base,cons,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Michel ten Voorde, Apr 14 2001

STATUS

approved

A068438

Expansion of Pi in base 13.

+10
21

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..92.

EXAMPLE

3.1ac1049052a2c77369c0aa89cc988327829835...

MATHEMATICA

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+ *)

CROSSREFS

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.

KEYWORD

easy,nonn,base,cons

AUTHOR

Benoit Cloitre, Mar 09 2002

STATUS

approved

A068439

Expansion of Pi in base 14.

+10
21

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..90.

EXAMPLE

3.1da75cda81375427a40abcb1bd47549c89bcb6...

MATHEMATICA

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

CROSSREFS

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).

KEYWORD

easy,nonn,base,cons

AUTHOR

Benoit Cloitre, Mar 09 2002

STATUS

approved

A102243

Expansion of Pi in golden base (i.e., in irrational base phi = (1+sqrt(5))/2) = A001622.

+10
7

1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

3

COMMENTS

George Bergman wrote his paper when he was 12. Mike Wallace interviewed him when Bergman was 14. - Robert G. Wilson v, Mar 14 2014

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 3..1002 (offset adapted by Georg Fischer, Jan 24 2019)

George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), pp. 98-110.

Chittaranjan Pardeshi, 100000 digits of Pi in golden base

Chittaranjan Pardeshi, BBP-Like formula for Pi in Golden Ratio Base Phi

Mike Wallace, Mike Wallace Asks George Bergman: What Makes a Genius Tick?, Math. Mag. 31 (1958), p. 282.

FORMULA

Pi = 4/phi + Sum_{n>=0} (1/phi^(12*n)) * ( 8/((12*n+3)*phi^3) + 4/((12*n+5)*phi^5) - 4/((12*n+7)*phi^7) - 8/((12*n+9)*phi^9) - 4/((12*n+11)*phi^11) + 4/((12*n+13)*phi^13) ) where phi = (1+sqrt(5))/2. - Chittaranjan Pardeshi, May 16 2022

EXAMPLE

Pi = phi^2 + 1/phi^2 + 1/phi^5 + 1/phi^7 + ... thus Pi = 100.0100101010010001010101000001010... in golden base.

MATHEMATICA

RealDigits[Pi, GoldenRatio, 111][[1]] (* Robert G. Wilson v, Feb 26 2010 *)

PROG

(PARI) f=(1+sqrt(5))/2; z=Pi; b=0; m=100; for(n=1, m, c=ceil(log(z)/log(1/f)); z=z-1/f^c; b=b+1./10^c; if(n==m, print1(b, ", ")))

(PARI)

alist(len) = {

my(phi=quadgen(5), n=-1, pi=4/phi, gap=phi^3, hi=pi+gap, t=0, w=phi^3);

vector(len, i,

w = w/phi;

while(t+w < hi && t+w > pi,

n = n + 1;

pi += phi^(-12*n) * (

8 * phi^-3 / (12*n+3)

+ 4 * phi^-5 / (12*n+5)

- 4 * phi^-7 / (12*n+7)

- 8 * phi^-9 / (12*n+9)

- 4 * phi^-11 / (12*n+11)

+ 4 * phi^-13 / (12*n+13));

gap /= phi^12;

hi = pi + gap);

if( t+w <= pi, t += w; 1, 0))};

alist(1000) \\ Chittaranjan Pardeshi, May 18 2022

CROSSREFS

Cf. A000796, A001622, A004601, A004602, A004603, A004604, A004605, A004606, A004608, A006941, A062964, A068436, A068437, A068438, A068439, A068440, A238897.

KEYWORD

cons,base,nonn

AUTHOR

Benoit Cloitre, Feb 18 2005

EXTENSIONS

Offset corrected by Lee A. Newberg, Apr 13 2018

STATUS

approved

A238897

Pi in the base sqrt(2).

+10
4

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

4

COMMENTS

George Bergman wrote his paper when he was 12. Mike Wallace interviewed him when Bergman was 14.

LINKS

T. D. Noe, Table of n, a(n) for n = 4..1007

George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), pp. 98-110.

Mike Wallace, Mike Wallace Asks George Bergman: What Makes a Genius Tick?, Math. Mag. 31 (1958), p. 282.

EXAMPLE

= 1000.00010001000000000000010010000000000100001000010000000010000001...

MATHEMATICA

RealDigits[Pi, Sqrt[2], 105][[1]]

CROSSREFS

Cf. A050948 (Pi in base e), A102243 (Pi in the golden base), A239199 (Pi in base sqrt(3)).

Cf. A000796, A004601, A004602, A004603, A004604, A004605, A004606, A004608, A006941, A062964, A068436, A068437, A068438, A068439, A068440.

KEYWORD

nonn,base

AUTHOR

T. D. Noe, Mar 11 2014

STATUS

approved

A278974

In the ternary Pi race between digits zero and one, where the race leader changes.

+10
4

1, 3, 8, 1481, 1505, 1509, 1513, 1541, 1567, 1596, 1730, 1734, 1739, 1741, 1769, 1772, 1783, 1790, 66446, 66489, 66493, 66496, 68547, 68554, 68871, 69116, 69146, 69190, 69194, 69268, 69270, 69379, 69381, 69389, 241170

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Hans Havermann and Robert G. Wilson v, Table of n, a(n) for n = 1..395

EXAMPLE

Ternary Pi is 10.01021101222201021100211...

With no digits of ternary Pi, there are an equal number of zeros and ones. 1 is in the sequence because with the initial digit of ternary Pi, 1 has now taken the count lead over 0 (1-0). 3 is the next term because with 3 initial digits of ternary Pi, 0 has now taken the count lead over 1 (2-1). 8 is the next term because with 8 initial digits, 1 regains the count lead over 0 (4-3).

MATHEMATICA

pib = RealDigits[Pi, 3, 5000000][[1]]; flag = 1; z = o = t = 0; k = 1; lst = {}; While[k < 5000001, Switch[ pib[[k]], 0, z++, 1, o++, 2, t++]; If[(z > o && flag != 1) || (z < o && flag != -1), AppendTo[lst, k]; flag = -flag]; k++]; lst

CROSSREFS

Cf. A004602, A278920, A278975, A278976, A278977.

KEYWORD

nonn,base

AUTHOR

Hans Havermann and Robert G. Wilson v, Dec 03 2016

STATUS

approved

A278975

In the ternary Pi race between digits zero and two, where the race leader changes.

+10
4

2, 14, 17, 33, 156, 189, 4853, 5494, 5541, 5548, 5663, 5665, 5668, 5673, 5686, 5689, 5702, 5704, 5719, 5732, 5739, 5831, 5834, 5839, 5845, 5847, 5905, 5913, 5925, 5928, 5950, 5978, 5980, 5986, 6000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Hans Havermann, Table of n, a(n) for n = 1..2508

EXAMPLE

Ternary Pi is 10.01021101222201021100211...

With no digits of ternary Pi, there are an equal number of zeros and twos. 2 is in the sequence because with the initial 2 digits of ternary Pi, 0 has now taken the count lead over 2 (1-0). 14 is the next term because with 14 initial digits of ternary Pi, 2 has now taken the count lead over 0 (5-4). 17 is the next term because with 17 initial digits, 0 regains the count lead over 2 (6-5).

MATHEMATICA

pib = RealDigits[Pi, 3, 5000000][[1]]; flag = -1; z = o = t = 0; k = 1; lst = {}; While[k < 5000001, Switch[ pib[[k]], 0, z++, 1, o++, 2, t++]; If[(z > t && flag != 1) || (z < t && flag != -1), AppendTo[lst, k]; flag = -flag]; k++]; lst

CROSSREFS

Cf. A004602, A278920, A278974, A278976, A278978.

KEYWORD

nonn,base

AUTHOR

Hans Havermann and Robert G. Wilson v, Dec 03 2016

STATUS

approved