A021017 -id:A021017 - cp's OEIS frontend

A021069 Decimal expansion of 1/65.

0, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5

Offset: 0

N. J. A. Sloane

Without the leading 0 also the decimal expansion of 2/13.

			0.0153846153846...  - _Natan Arie Consigli_, Sep 18 2016

G. C. Greubel, Table of n, a(n) for n = 0..10000
Christian Táfula, Knights are 24/13 times faster than the king, arXiv:2405.19589 [math.CO], 2024.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A010734, A020806, A021017, A021095, A128834, A130806, A144472.

Magma
```
1./65; // G. C. Greubel, Jan 15 2018
```

Digits:=100: evalf(1/65); # Wesley Ivan Hurt, Sep 20 2016

Join[ConstantArray[0, Abs@ Last@ #], First@ #] &@ RealDigits[N[1/65, 120]] (* Michael De Vlieger, Sep 06 2016 *)

1./65. /* Michel Marcus, Aug 22 2016 */

Equals 2 - 24/13. See Táfula link. - Michel Marcus, May 31 2024

G.f.: x*(1 + 4*x - 2*x^2 + 6*x^3)/((1 - x)*(1 + x)*(1 - x + x^2)). - Stefano Spezia, Apr 30 2025

A091722 Babylonian sexagesimal (base 60) expansion of 1/13.

Original entry on oeis.org

4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55, 23, 4, 36, 55

Offset: 0

Jeppe Stig Nielsen, Feb 01 2004

Index entries for linear recurrences with constant coefficients, signature (1, -1, 1).

Cf. A021017, A091721, A091722, A055643.

Cf. A159991. - Reinhard Zumkeller, May 02 2009

RealDigits[ 1/13, 60, 75] [[1]] (* Robert G. Wilson v, Feb 02 2004 *)

A242826 Primes formed by the initial digits of the decimal expansion of 1/13, starting at the first nonzero digit in the expansion.

Original entry on oeis.org

7, 769, 769230769, 769230769230769230769, 769230769230769230769230769230769

Offset: 1

Felix Fröhlich, May 23 2014

Cf. A021017.

Corresponding sequences for 1/k: A242824 (k=7), A093676 (k=12), A242827 (k=14), A242828 (k=17), A242833 (k=19).

A368476 Decimal expansion of 109/65, being the highest possible win/loss points ratio in a completed 3-set tennis match, with 10-point final tie-break, which the player loses.

Original entry on oeis.org

1, 6, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7, 6, 9, 2, 3, 0, 7

Offset: 1

Marco Ripà, Dec 26 2023

Tie-break games are played to 7 points in all sets except the last of the match, which is a 10-point tie-break.
The structure of sets and games in tennis means a player can win more points but lose the match.
The highest win/loss ratio for 3 sets occurs with game scores 6-0 6-7 6-7, where player A wins games by points score 4-0, and loses by 2-4 in ordinary games and 5-7 8-10 in the two tie-break games.
Player A wins 109 points and player B wins 65 points, but player A loses the match.
This ratio is a little lower than when the final tie-break is played to 7 points (see A368009).

			1.6769230... (periodic part 769230).

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A368008, A368009, A368146.

Apart from leading digits the same as A021017.

First[RealDigits[109/65, 10, 100]] (* or *)
PadRight[{1, 6}, 100, {3, 0, 7, 6, 9, 2}] (* Paolo Xausa, Jan 30 2024 *)

Equals (6*4 + (6*4 + 6*2 + 5) + (6*4 + 6*2 + 8))/((6*4 + 7) + (6*4 + 10)).

A219705 Decimal expansion of cos(log(2)).

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Nov 25 2012

In a letter to Christian Goldbach dated December 9, 1741, Leonhard Euler gave 10/13 as a rational approximation of this number.
Also, real part of 2^i. - Bruno Berselli, Dec 31 2012
The imaginary part of 2^i is A220085. - Robert G. Wilson v, Feb 04 2013

			0.76923890136...

Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 309.
W. Michael Kelley, The Humongous Book of Calculus Problems. New York: Alpha Books (Penguin Group) p. 233, Problem 15.22.

Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton, New Jersey: Princeton University Press (1988), 143 - 144.
Elizabeth Volz, An English translation of portions of seven correspondences between Euler and Goldbach on Euler’s complex exponential paradox and special values of cosine, 2008
Elizabeth Volz and Hieu D. Nguyen, Euler, Goldbach and exact values of trigonometric functions, 2008 preprint

Cf. A002162, A021017, A220085 (imaginary part of 2^i).

Mathematica
```
RealDigits[Cos[Log[2]], 10, 105][[1]]
```

fpprec:110; ev(bfloat(cos(log(2)))); /* Bruno Berselli, Dec 31 2012 */

cos(log(2)) \\ Charles R Greathouse IV, Nov 25 2012

cos(log(2)) = (2^i + 2^(-i))/2.

a(43) ff. corrected by Georg Fischer, Apr 03 2020

A355068 Square array read by upwards antidiagonals: T(n,k) = k-th digit after the decimal point in decimal expansion of 1/n, for n >= 1 and k >= 1.

Original entry on oeis.org

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

Offset: 1

Chittaranjan Pardeshi, Jun 17 2022

First row is all zeros since n=1 has all zeros after the decimal point.

			Array begins:
      k=1  2  3  4  5  6  7  8
  n=1:  0, 0, 0, 0, 0, 0, 0, 0,
  n=2:  5, 0, 0, 0, 0, 0, 0, 0,
  n=3:  3, 3, 3, 3, 3, 3, 3, 3,
  n=4:  2, 5, 0, 0, 0, 0, 0, 0,
  n=5:  2, 0, 0, 0, 0, 0, 0, 0,
  n=6:  1, 6, 6, 6, 6, 6, 6, 6,
  n=7:  1, 4, 2, 8, 5, 7, 1, 4,
  n=8:  1, 2, 5, 0, 0, 0, 0, 0,
Row n=7 is 1/7 = .142857142857..., whose digits after the decimal point are 1,4,2,8,5,7,1,4,2,8,5,7, ...

Cf. A020806, A021017, A021018.
Cf. A048962, A257927.
Cf. A061480 (diagonal).
Cf. A355202 (binary).

T(n,k) = my(r=lift(Mod(10,n)^(k-1))); floor(10*r/n)%10;

def T(n,k): return (10*pow(10,k-1,n)//n)%10

1/n = Sum_{k>=1} T(n, k)*10^-k, for n > 1.

A021108 Decimal expansion of 1/104.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

After 9, periodic with period 6: [6, 1, 5, 3, 8, 4]. See also A021030 (1/26), A021069 (1/65), A021420 (1/416), A021654 (1/650). - Bruno Berselli, Apr 13 2018

			0.009615384615384615384615384615384615384615384615384615384615384...

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Join[{0, 0}, RealDigits[1/104, 10, 120][[1]]] (* or *) PadRight[{0, 0, 9, 6}, 120,{3, 8, 4, 6, 1, 5}] (* Harvey P. Dale, Aug 18 2012 *)

Equals A020821 * A021017 = A020773 * A021030 = A020761 * A021056. - Bruno Berselli, Apr 13 2018

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A021069 Decimal expansion of 1/65.

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A091722 Babylonian sexagesimal (base 60) expansion of 1/13.

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A242826 Primes formed by the initial digits of the decimal expansion of 1/13, starting at the first nonzero digit in the expansion.

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A368476 Decimal expansion of 109/65, being the highest possible win/loss points ratio in a completed 3-set tennis match, with 10-point final tie-break, which the player loses.

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A219705 Decimal expansion of cos(log(2)).

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A355068 Square array read by upwards antidiagonals: T(n,k) = k-th digit after the decimal point in decimal expansion of 1/n, for n >= 1 and k >= 1.

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A021108 Decimal expansion of 1/104.

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