A374334 -id:A374334 - cp's OEIS frontend

A062964 Pi in hexadecimal.

3, 2, 4, 3, 15, 6, 10, 8, 8, 8, 5, 10, 3, 0, 8, 13, 3, 1, 3, 1, 9, 8, 10, 2, 14, 0, 3, 7, 0, 7, 3, 4, 4, 10, 4, 0, 9, 3, 8, 2, 2, 2, 9, 9, 15, 3, 1, 13, 0, 0, 8, 2, 14, 15, 10, 9, 8, 14, 12, 4, 14, 6, 12, 8, 9, 4, 5, 2, 8, 2, 1, 14, 6, 3, 8, 13, 0, 1, 3, 7, 7, 11, 14, 5, 4, 6, 6, 12, 15, 3, 4, 14, 9

Offset: 1

Robert Lozyniak (11(AT)onna.com), Jul 22 2001

Bailey and Crandall conjecture that the terms of this sequence, apart from the first, are given by the formula floor(16*(x(n) - floor(x(n)))), where x(n) is determined by the recurrence equation x(n) = 16*x(n-1) + (120*n^2 - 89*n + 16)/(512*n^4 - 1024*n^3 + 712*n^2 - 206*n + 21) with the initial condition x(0) = 0 (see A374334). They have numerically verified the conjecture for the first 100000 terms of the sequence. - Peter Bala, Oct 31 2013

Bailey, Borwein & Plouffe's ("BBP") formula allows one to compute the n-th hexadecimal digit of Pi without calculating the preceding digits (see Wikipedia link). - M. F. Hasler, Mar 14 2015

			3.243f6a8885a308d3...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 17-28.

Harry J. Smith, Table of n, a(n) for n = 1..20000
D. H. Bailey, Compendium of BBP-Type Formulas for Mathematical Constants.
D. H. Bailey and R. E. Crandall, On the Random Character of Fundamental Constant Expansions, Experiment. Math. Volume 10, Issue 2 (2001), 175-190.
CalcCrypto, Pi in Hexadecimal. [Broken link]
Steven R. Finch, The Miraculous Bailey-Borwein-Plouffe Pi Algorithm.
Steve Pagliarulo, Stu's pi page: base 16 (31 pages of numbers). [Dead link]
Johnny Vogler, More digits.
Wikipedia, Bailey-Borwein-Plouffe formula.

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

Cf. A007514, A374334.

Mathematica
```
RealDigits[ N[ Pi, 115], 16] [[1]]
```

{ default(realprecision, 24300); x=Pi; for (n=1, 20000, d=floor(x); x=(x-d)*16; write("b062964.txt", n, " ", d)); } \\ Harry J. Smith, Apr 27 2009

N=50; default(realprecision,.75*N); A062964=digits(Pi*16^N\1,16) \\ M. F. Hasler, Mar 14 2015

a(n) = 8*A004601(4n) + 4*A004601(4n+1) + 2*A004601(4n+2) + 1*A004601(4n+3).

If Pi is the expansion of Pi in base 10, Pi=3.1415926...: a(n) = floor(16^n*Pi) - 16*floor(16^(n-1)*Pi). - Benoit Cloitre, Mar 09 2002

More terms from Henry Bottomley, Jul 24 2001

A374335 a(n) is the denominator of x(n) = (16x(n-1) + (120n^2 - 89n + 16)/(512n^4 - 1024n^3 + 712n^2 - 206*n + 21)) mod 1, with x(0) = 0.

Original entry on oeis.org

1, 15, 4095, 765765, 111035925, 78058255275, 24536311574775, 81926744348173725, 154923473562396513975, 154923473562396513975, 595232293160786606325, 76784965817741472215925, 321191512015612578279214275, 3146713243216956429401462252175, 342991743510648250804759385487075

Offset: 0

Paolo Xausa, Jul 06 2024

See A374334 for details and links.

Paolo Xausa, Table of n, a(n) for n = 0..375

Cf. A374333, A374334 (numerators), A374581, A374608.

Block[{n = 0}, Denominator[NestList[Mod[16*# + (120*(++n)^2 - 89*n + 16)/(512*n^4 - 1024*n^3 + 712*n^2 - 206*n + 21), 1] &, 0, 20]]]

A374332 a(n) is the numerator of x(n) = (2*x(n-1) + 1/n) mod 1, with x(0) = 0.

Original entry on oeis.org

0, 0, 1, 1, 11, 1, 7, 64, 289, 1007, 44, 338, 163, 3505, 8297, 44488, 27221, 823117, 993287, 20403983, 26327699, 27713369, 27650353, 315868349, 2488325579, 6016553239, 1399433807, 3562923992, 9142117861, 275160597119, 268889538733, 3968532770473, 114095155444597

Offset: 0

Paolo Xausa, Jul 06 2024

A constant alpha, defined as alpha = Sum_{n >= 1} p(n)/(q(n)*b^n), is b-normal if and only if the associated sequence, defined by x(0) = 0 and x(n) = (b*x(n-1) + p(n)/q(n)) mod 1, is equidistributed in the unit interval.
The present sequence gives the numerators of the associated sequence for alpha = log(2) (where b = 2). See Bailey and Borwein (2005), p. 505 (first example of Theorem 3).
Denominators are given by A374333.

Paolo Xausa, Table of n, a(n) for n = 0..2000
David H. Bailey and Jonathan M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the American Mathematical Society, May 2005, Vol. 52, No. 5, pp. 502-514.
David H. Bailey and Richard E. Crandall, On the Random Character of Fundamental Constant Expansions, Experimental Mathematics, Vol. 10 (2001), Issue 2, pp. 175-190 (preprint draft).
David H. Bailey and Richard E. Crandall, Random Generators and Normal Numbers, Experimental Mathematics, Vol. 11 (2002), Issue 4, pp. 527-546 (preprint draft).

Cf. A002162, A374333 (denominators), A374334, A374336.

Block[{n = 0}, Numerator[NestList[Mod[2*# + 1/++n, 1] &, 0, 50]]]

x(n) = if (n==0, 0, 2*x(n-1) + 1/n);
a(n) = numerator(frac(x(n))); \\ Michel Marcus, Jul 13 2024

from fractions import Fraction
from itertools import count, islice
def A374332_gen(): # generator of terms
    a = Fraction(0,1)
    for n in count(1):
        yield a.numerator
        a = (2*a+Fraction(1,n)) % 1
A374332_list = list(islice(A374332_gen(),20)) # Chai Wah Wu, Jul 13 2024

A374580 a(n) is the numerator of (120n^2 + 151n + 47)/(512n^4 + 1024n^3 + 712n^2 + 194n + 15).

Original entry on oeis.org

47, 106, 829, 316, 857, 3802, 5273, 776, 1787, 11126, 4519, 16228, 19139, 1486, 25681, 29312, 3687, 37294, 8329, 15412, 51067, 56138, 20483, 2680, 72791, 8758, 85093, 91604, 6557, 105346, 112577, 40016, 127759, 27142, 15989, 152332, 161003, 56638, 35813, 188456

Offset: 0

Paolo Xausa, Jul 12 2024

See Bailey and Crandall (2001), section 5 (pp. 183-184) for a derivation of this rational polynomial.

Denominators are given by A374581.

Paolo Xausa, Table of n, a(n) for n = 0..10000
David H. Bailey and Richard E. Crandall, On the Random Character of Fundamental Constant Expansions, Experimental Mathematics, Vol. 10 (2001), Issue 2, pp. 175-190 (preprint draft).

Cf. A000796, A001025, A374334, A374581 (denominators), A374607.

A374580[n_] := Numerator[(120*n^2 + 151*n + 47)/(512*n^4 + 1024*n^3 + 712*n^2 + 194*n + 15)];
Array[A374580, 50, 0]

from math import gcd
def A374580(n): return (lambda p,q: p//gcd(p,q))(n*(120*n + 151) + 47,n*(n*(n*(512*n + 1024) + 712) + 194) + 15) # Chai Wah Wu, Jul 14 2024

Sum_{n >= 0} (1/16^n)*a(n)/A374581(n) = A000796. See Bailey and Crandall (2001), eq. 5-2, p. 184.

A374607 a(n) is the numerator of (1134n^3 + 2097n^2 + 1188n + 193)/(10368n^4 + 20736n^3 + 14112n^2 + 3744*n + 320).

Original entry on oeis.org

193, 1153, 20029, 832, 111073, 50077, 327757, 7816, 724513, 251857, 1355773, 55511, 2275969, 715357, 3539533, 134909, 5200897, 1549441, 7314493, 133717, 9934753, 2862973, 13116109, 233347, 16912993, 4764817, 21379837, 746297, 26571073, 7363837, 32541133, 1119851

Offset: 0

Paolo Xausa, Jul 13 2024

See Bailey and Crandall (2001), section 5 (pp. 183-185) for a derivation of this rational polynomial.

Denominators are given by A374608.

Paolo Xausa, Table of n, a(n) for n = 0..10000
David H. Bailey and Richard E. Crandall, On the Random Character of Fundamental Constant Expansions, Experimental Mathematics, Vol. 10 (2001), Issue 2, pp. 175-190 (preprint draft).

Cf. A000796, A010482, A089357, A374334, A374580, A374608 (denominators).

A374607[n_] := Numerator[(1134*n^3 + 2097*n^2 + 1188*n + 193)/(10368*n^4 + 20736*n^3 + 14112*n^2 + 3744*n + 320)];
Array[A374607, 50, 0]

from math import gcd
def A374607(n): return (p:=n*(n*(1134*n + 2097) + 1188) + 193)//gcd(p,n*(n*(n*(324*n + 648) + 441) + 117) + 10<<5) # Chai Wah Wu, Jul 14 2024

sqrt(27)*(Sum_{n >= 0} (1/64^n)*a(n)/A374608(n)) = A000796. See Bailey and Crandall (2001), p. 185.

A374336 a(n) is the numerator of x(n) = (2*x(n-1) + c(n)) mod 1, where c(n) = 1/n if n is a power of 3 and 0 otherwise, with x(0) = 0.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 2, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 13, 26, 25, 23, 19, 11, 22, 17, 7, 14, 1, 2, 4, 8, 16, 5, 10, 20, 13, 26, 25, 23, 19, 11, 22, 17, 7, 14, 1, 2, 4, 8, 16, 5, 10, 20, 13, 26, 25, 23, 19, 11, 22, 17, 7, 14, 1, 2, 4

Offset: 0

Paolo Xausa, Jul 06 2024

A constant alpha, defined as alpha = Sum_{n >= 1} p(n)/(q(n)*b^n), is b-normal if and only if the associated sequence, defined by x(0) = 0 and x(n) = (b*x(n-1) + p(n)/q(n)) mod 1, is equidistributed in the unit interval.
The present sequence gives the numerators of the associated sequence (where b = 2) for alpha_0 = Sum_{n >= 1} 1/((3^n)*2^(3^n)) = A192014. See Bailey and Borwein (2005), pp. 505-506 (third example of Theorem 3). They show that alpha_0, as well as any constant defined as Sum_{n >= 1} 1/((3^n)*2^(3^n+r_n)) (where r_n is the n-th binary digit of the real number r in the [0,1) interval), is 2-normal and transcendental.
Bailey and Borwein also note that terms follow a pattern of triply repeating segments, each of length 2*3^m and containing all integers relative prime to and less than 3^(m+1).
Denominators are given by A365458.

Paolo Xausa, Table of n, a(n) for n = 0..10000
David H. Bailey and Jonathan M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the American Mathematical Society, May 2005, Vol. 52, No. 5, pp. 502-514.

Cf. A192014, A374332, A374334, A365458 (denominators).

Block[{n = 0}, Numerator[NestList[Mod[2*# + If[IntegerQ[Log[3, ++n]], 1/n, 0], 1] &, 0, 100]]]

cp's OEIS Frontend

A062964 Pi in hexadecimal.

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A374335 a(n) is the denominator of x(n) = (16*x(n-1) + (120*n^2 - 89*n + 16)/(512*n^4 - 1024*n^3 + 712*n^2 - 206*n + 21)) mod 1, with x(0) = 0.

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A374332 a(n) is the numerator of x(n) = (2*x(n-1) + 1/n) mod 1, with x(0) = 0.

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A374580 a(n) is the numerator of (120*n^2 + 151*n + 47)/(512*n^4 + 1024*n^3 + 712*n^2 + 194*n + 15).

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A374607 a(n) is the numerator of (1134*n^3 + 2097*n^2 + 1188*n + 193)/(10368*n^4 + 20736*n^3 + 14112*n^2 + 3744*n + 320).

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Programs

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Python

Formula

A374336 a(n) is the numerator of x(n) = (2*x(n-1) + c(n)) mod 1, where c(n) = 1/n if n is a power of 3 and 0 otherwise, with x(0) = 0.

Views

Author

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Comments

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Programs

Mathematica

A374335 a(n) is the denominator of x(n) = (16x(n-1) + (120n^2 - 89n + 16)/(512n^4 - 1024n^3 + 712n^2 - 206*n + 21)) mod 1, with x(0) = 0.

A374580 a(n) is the numerator of (120n^2 + 151n + 47)/(512n^4 + 1024n^3 + 712n^2 + 194n + 15).

A374607 a(n) is the numerator of (1134n^3 + 2097n^2 + 1188n + 193)/(10368n^4 + 20736n^3 + 14112n^2 + 3744*n + 320).