A374332 -id:A374332 - cp's OEIS frontend

A374334 a(n) is the numerator 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.

0, 2, 1076, 188663, 106894973, 32442016954, 16143697977964, 43667396600461261, 82482175187690988496, 80845733759021750791, 209616749220518838502, 48891577015658186678698, 60882892596227901210360094, 108196850082040258114673507582, 189145139720511629801253759599798

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 (with b = 2) for alpha = Pi. See Bailey and Borwein (2005), p. 505 (second example of Theorem 3). In the same paper, on p. 513, they conjecture that, for n >= 1, y(n) = floor(16*x(n)) = A062964(n+1). See also Bailey and Crandall (2001), p. 176.
Denominators are given by A374335.

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

Cf. A000796, A062964, A374332, A374335 (denominators), A374336, A374580, A374607.

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

A374333 a(n) is the denominator of x(n) = (2*x(n-1) + 1/n) mod 1, with x(0) = 0.

Original entry on oeis.org

1, 1, 2, 3, 12, 30, 30, 105, 840, 1260, 63, 693, 2772, 18018, 18018, 45045, 720720, 6126120, 3063060, 29099070, 58198140, 29099070, 29099070, 334639305, 2677114440, 6692786100, 1673196525, 5019589575, 20078358300, 291136195350, 291136195350, 4512611027925, 144403552893600

Offset: 0

Paolo Xausa, Jul 06 2024

See A374332 for details and links.

Paolo Xausa, Table of n, a(n) for n = 0..2000

Cf. A374332 (numerators), A374335.

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

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

from itertools import count, islice
from fractions import Fraction
def A374333_gen(): # generator of terms
    a = Fraction(0,1)
    for n in count(1):
        yield a.denominator
        a = (2*a+Fraction(1,n)) % 1
A374333_list = list(islice(A374333_gen(),20)) # Chai Wah Wu, Jul 13 2024

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

A374334 a(n) is the numerator 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.

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A374333 a(n) is the denominator of x(n) = (2*x(n-1) + 1/n) mod 1, with x(0) = 0.

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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.

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cp's OEIS Frontend

A374334 a(n) is the numerator 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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A374333 a(n) is the denominator of x(n) = (2*x(n-1) + 1/n) mod 1, with x(0) = 0.

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Crossrefs

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Mathematica

PARI

Python

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

A374334 a(n) is the numerator 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.