A374580 -id:A374580 - 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]]]

A374581 a(n) is the denominator of (120n^2 + 151n + 47)/(512n^4 + 1024n^3 + 712n^2 + 194n + 15).

Original entry on oeis.org

15, 819, 19635, 15225, 69597, 466785, 911547, 179645, 533715, 4165161, 2072385, 8947437, 12491175, 1133055, 22621131, 29539125, 4214903, 48002745, 11990775, 24669567, 90400695, 109375617, 43730505, 6244749, 184439871, 24049985, 252455907, 292777485, 22516425, 387706641

Offset: 0

Paolo Xausa, Jul 12 2024

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

Numerators are given by A374580.

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, A374335, A374580 (numerators), A374608.

A374581[n_] := Denominator[(120*n^2 + 151*n + 47)/(512*n^4 + 1024*n^3 + 712*n^2 + 194*n + 15)];
Array[A374581, 30, 0]

from math import gcd
def A374581(n): return (lambda p,q: q//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)*A374580(n)/a(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.

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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A374581 a(n) is the denominator of (120n^2 + 151n + 47)/(512n^4 + 1024n^3 + 712n^2 + 194n + 15).

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

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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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A374581 a(n) is the denominator of (120*n^2 + 151*n + 47)/(512*n^4 + 1024*n^3 + 712*n^2 + 194*n + 15).

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Formula

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

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.

A374581 a(n) is the denominator 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).