A374608 -id:A374608 - cp's OEIS frontend

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.

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]]]

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

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.

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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

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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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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Python

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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Author

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Programs

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Python

Formula

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.

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).