A010158 -id:A010158 - cp's OEIS frontend

A041150 Numerators of continued fraction convergents to sqrt(85).

9, 37, 46, 83, 378, 6887, 27926, 34813, 62739, 285769, 5206581, 21112093, 26318674, 47430767, 216041742, 3936182123, 15960770234, 19896952357, 35857722591, 163327842721, 2975758891569, 12066363408997

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 17 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A087798.
For the terms of the periodic sequence of the continued fraction for sqrt(85) see A010158. We observe that its period is five. The decimal expansion of sqrt(85) is A010536. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,756,0,0,0,0,1).

Cf. A010536, A041018, A041046, A041090, A041150, A041151, A041226, A041318, A041426,

A041550.

Numerator[Convergents[Sqrt[85], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

From Johannes W. Meijer, Jun 17 2010: (Start)
a(5*n) = A087798(3*n+1), a(5*n+1) = (A087798(3*n+2) - A087798(3*n+1))/2, a(5*n+2) = (A087798(3*n+2) + A087798(3*n+1))/2, a(5*n+3) = A087798(3*n+2) and a(5*n+4) = A087798(3*n+3)/2. (End)
G.f.: -(x^9-9*x^8+37*x^7-46*x^6+83*x^5+378*x^4+83*x^3+46*x^2+37*x+9) / (x^10+756*x^5-1). - Colin Barker, Nov 04 2013

A041151 Denominators of continued fraction convergents to sqrt(85).

Original entry on oeis.org

1, 4, 5, 9, 41, 747, 3029, 3776, 6805, 30996, 564733, 2289928, 2854661, 5144589, 23433017, 426938895, 1731188597, 2158127492, 3889316089, 17715391848, 322766369353, 1308780869260, 1631547238613, 2940328107873, 13392859670105, 244011802169763, 989440068349157

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 12 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A099371.
For the terms of the periodic sequence of the continued fraction for sqrt(85) see A010158. We observe that its period is five. The decimal expansion of sqrt(85) is A010536. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,756,0,0,0,0,1).

Cf. A041150, A041019, A041047, A041091, A041151, A041227, A041319, A041427, A041551.

I:=[1, 4, 5, 9, 41, 747, 3029, 3776, 6805, 30996]; [n le 10 select I[n] else 756*Self(n-5)+Self(n-10): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[85], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[85], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5*n) = A099371(3*n+1), a(5*n+1) = (A099371(3*n+2)-A099371(3*n+1))/2, a(5*n+2) = (A099371(3*n+2)+A099371(3*n+1))/2, a(5*n+3):= A099371(3*n+2) and a(5*n+4) = A099371(3*n+3)/2. (End)
G.f.: -(x^8-4*x^7+5*x^6-9*x^5+41*x^4+9*x^3+5*x^2+4*x+1) / (x^10+756*x^5-1). - Colin Barker, Nov 11 2013
a(n) = 756*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 12 2013

A010536 Decimal expansion of square root of 85.

Original entry on oeis.org

9, 2, 1, 9, 5, 4, 4, 4, 5, 7, 2, 9, 2, 8, 8, 7, 3, 1, 0, 0, 0, 2, 2, 7, 4, 2, 8, 1, 7, 6, 2, 7, 9, 3, 1, 5, 7, 2, 4, 6, 8, 0, 5, 0, 4, 8, 7, 2, 2, 4, 6, 4, 0, 0, 8, 0, 0, 7, 7, 5, 2, 2, 0, 5, 4, 4, 2, 6, 7, 1, 0, 2, 6, 8, 0, 1, 8, 7, 5, 4, 6, 0, 7, 6, 7, 8, 9, 4, 0, 9, 0, 7, 9, 3, 2, 8, 0, 5, 6

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 9 followed by {4, 1, 1, 4, 18} repeated. - Harry J. Smith, Jun 10 2009

			9.219544457292887310002274281762793157246805048722464008007752205442671....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

Cf. A010158 (continued fraction). - Harry J. Smith, Jun 10 2009

RealDigits[N[85^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 23 2012 *)
RealDigits[Sqrt[85],10,120][[1]] (* Harvey P. Dale, Jul 23 2024 *)

default(realprecision, 20080); x=sqrt(85); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010536.txt", n, " ", d));  \\ Harry J. Smith, Jun 10 2009

from math import isqrt
def aupton(nn): return list(map(int, str(isqrt(85 * 10**(2*nn)))))[:nn]
print(aupton(100)) # Michael S. Branicky, Sep 02 2021

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A041150 Numerators of continued fraction convergents to sqrt(85).

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A041151 Denominators of continued fraction convergents to sqrt(85).

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A010536 Decimal expansion of square root of 85.

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