A010467 -id:A010467 - cp's OEIS frontend

A004585 Expansion of sqrt(10) in base 2.

1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1

Offset: 2

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 2..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010467.

d:= 10; m:=2; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 25 2018

RealDigits[Sqrt[10], 2, 100][[1]] (* G. C. Greubel, Mar 25 2018 *)

my(b = binary(sqrt(10))); concat(b[1], b[2]) \\ Michel Marcus, Mar 26 2018

A348670 Decimal expansion of 10 - Pi^2.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Oct 29 2021

Let ABC be a unit-area triangle, and let P be a point uniformly picked at random inside it. Let D, E and F be the intersection points of the lines AP, BP and CP with the sides BC, CA and AB, respectively. Then, the expected value of the area of the triangle DEF is this constant.

			0.13039559891064138116550900012384886468630059275920...

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013, p. 220.
A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, p. 275, ex. 2.5.3.

R. W. Gosper, Acceleration of Series, AIM-304 (1974), page 71.
Olivier Schneegans, How Close to 10 is Pi^2?, The American Mathematical Monthly, Vol. 126, No. 5 (2019), p. 448.
Daniel Sitaru, Problem B131, Crux Mathematicorum, Vol. 49, No. 7 (2023), p. 381.
Index entries for transcendental numbers.

Cf. A002388, A008911, A010467, A060459.

Mathematica
```
RealDigits[10 - Pi^2, 10, 100][[1]]
```
PARI
```
10 - Pi^2 \\ Michel Marcus, Oct 29 2021
```

Equals Sum_{k>=1} 1/(k*(k+1))^3 = Sum_{k>=1} 1/A060459(k).
Equals 6 * Sum_{k>=2} 1/(k*(k+1)^2*(k+2)) = Sum_{k>=3} 1/A008911(k).
Equals 2 * Integral_{x=0..1, y=0..1} x*(1-x)*y*(1-y)/(1-x*y)^2 dx dy.
Equals 4 * Sum_{m,n>=1} (m-n)^2/(m*n*(m+1)^2*(n+1)^2*(m+2)*(n+2)) (Sitaru, 2023). - Amiram Eldar, Aug 18 2023

A004586 Expansion of sqrt(10) in base 3.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 0, 1, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 2, 1, 2, 0, 1, 2, 0, 1, 0, 0, 1, 0, 2, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 0, 2, 0, 0, 2, 2, 2, 0, 2, 0, 0, 2, 1, 0, 1, 0, 0, 2, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 1, 0, 0, 2, 1, 2, 0, 0, 1, 2, 0, 1, 1, 1

Offset: 2

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 2..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010467.

d:= 10; m:=3; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 25 2018

RealDigits[Sqrt[10], 3, 100][[1]] (* G. C. Greubel, Mar 25 2018 *)

A004587 Expansion of sqrt(10) in base 4.

Original entry on oeis.org

3, 0, 2, 2, 1, 2, 0, 2, 3, 0, 0, 1, 3, 1, 1, 2, 3, 1, 0, 2, 3, 1, 2, 2, 2, 1, 1, 0, 2, 1, 0, 0, 0, 2, 1, 1, 0, 1, 1, 1, 3, 2, 1, 0, 0, 1, 2, 0, 1, 2, 1, 2, 3, 0, 3, 1, 3, 3, 3, 3, 1, 1, 0, 2, 2, 2, 3, 1, 0, 3, 2, 0, 0, 2, 2, 2, 3, 2, 2, 3, 3, 2, 0, 1, 2, 0, 0, 0, 0, 0, 1, 3, 0, 1, 1, 1, 3, 1, 1

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010467.

d:= 10; m:=4; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 25 2018

RealDigits[Sqrt[10], 4, 100][[1]] (* G. C. Greubel, Mar 25 2018 *)

A004588 Expansion of sqrt(10) in base 5.

Original entry on oeis.org

3, 0, 4, 0, 1, 2, 0, 2, 4, 3, 2, 3, 4, 1, 4, 2, 0, 3, 1, 1, 3, 2, 2, 1, 1, 2, 2, 0, 1, 2, 0, 0, 3, 2, 3, 4, 4, 4, 4, 4, 2, 3, 4, 3, 3, 2, 4, 4, 2, 4, 0, 4, 4, 1, 4, 3, 0, 3, 1, 4, 4, 1, 2, 3, 3, 1, 3, 3, 4, 2, 4, 2, 2, 2, 0, 4, 0, 3, 3, 3, 4, 1, 2, 1, 4, 2, 4, 3, 3, 0, 2, 3, 4, 4, 1, 4, 0, 3, 1

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010467.

Prune(Reverse(IntegerToSequence(Isqrt(10*5^400), 5))); // Vincenzo Librandi, Jan 23 2016

RealDigits[Sqrt[10], 5, 100][[1]] (* Vincenzo Librandi, Jan 23 2016 *)

a(87) onwards corrected by Sean A. Irvine, Jan 22 2016

A004589 Expansion of sqrt(10) in base 6.

Original entry on oeis.org

3, 0, 5, 5, 0, 1, 5, 1, 2, 0, 5, 3, 2, 1, 0, 3, 1, 4, 4, 3, 1, 3, 0, 2, 5, 4, 4, 5, 1, 2, 0, 1, 3, 3, 1, 0, 1, 1, 4, 3, 0, 2, 2, 4, 0, 0, 0, 2, 0, 5, 1, 4, 4, 2, 1, 0, 4, 1, 5, 4, 0, 4, 5, 2, 1, 5, 2, 0, 1, 4, 0, 0, 1, 3, 0, 2, 2, 5, 5, 1, 3, 3, 1, 5, 2, 3, 1, 0, 5, 2, 0, 0, 0, 4, 4, 3, 5, 1, 2

Offset: 1

N. J. A. Sloane

			3.0550151205321...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010467.

Prune(Reverse(IntegerToSequence(Isqrt(10*6^400), 6))); // Vincenzo Librandi, Jan 23 2016

RealDigits[Sqrt[10], 6, 108][[1]] (* Alonso del Arte, Jul 31 2014 *)

a(79) onwards corrected by Sean A. Irvine, Jan 22 2016

A004590 Expansion of sqrt(10) in base 7.

Original entry on oeis.org

3, 1, 0, 6, 4, 4, 2, 5, 4, 2, 6, 3, 1, 4, 6, 1, 0, 0, 2, 5, 1, 4, 1, 5, 3, 6, 5, 3, 4, 3, 4, 6, 2, 1, 6, 3, 2, 5, 4, 0, 1, 6, 5, 2, 1, 0, 2, 1, 5, 6, 0, 6, 6, 4, 2, 6, 1, 6, 1, 0, 2, 6, 3, 2, 5, 6, 5, 4, 3, 1, 2, 1, 1, 6, 5, 5, 5, 2, 1, 3, 4, 0, 6, 5, 0, 6, 3, 2, 0, 5, 1, 1, 0, 0, 5, 1, 6, 0, 3

Offset: 1

N. J. A. Sloane

			3.106442542631461...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010467.

Prune(Reverse(IntegerToSequence(Isqrt(10*7^200),7))); // Jason Kimberley, Feb 19 2012

RealDigits[Sqrt[10], 7, 105][[1]] (* Alonso del Arte, Jul 31 2014 *)

a(72) onwards corrected by Sean A. Irvine, Jan 29 2016

A004591 Expansion of sqrt(10) in base 8.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010467.

Prune(Reverse(IntegerToSequence(Isqrt(10*8^200),8))); // Jason Kimberley, Feb 19 2012

RealDigits[Sqrt[10],8,120][[1]] (* Harvey P. Dale, Feb 04 2012 *)

More terms from Jason Kimberley, Feb 27 2012

A004592 Expansion of sqrt(10) in base 9.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010467.

Prune(Reverse(IntegerToSequence(Isqrt(10*9^200),9))); // Jason Kimberley, Feb 19 2012

RealDigits[Sqrt[10],9,100][[1]]  (* Harvey P. Dale, Mar 06 2011 *)

Corrected and extended by Harvey P. Dale, Mar 06 2011

More terms from Jason Kimberley, Feb 27 2012

A059177 Engel expansion of sqrt(10) = 3.16227...

Original entry on oeis.org

1, 1, 1, 7, 8, 12, 20, 86, 94, 118, 160, 179, 287, 315, 22588, 49419, 66011, 80779, 651661, 1078390, 1093865, 4254100, 27153191, 108815387, 220864645, 798937058, 992296124, 2196903274, 17524412379, 22828187385

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A010467.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[Sqrt[10], 7!], 10] (* modified by G. C. Greubel, Dec 26 2016 *)

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A004585 Expansion of sqrt(10) in base 2.

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A348670 Decimal expansion of 10 - Pi^2.

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A004586 Expansion of sqrt(10) in base 3.

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A004587 Expansion of sqrt(10) in base 4.

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A004588 Expansion of sqrt(10) in base 5.

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A004589 Expansion of sqrt(10) in base 6.

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A004590 Expansion of sqrt(10) in base 7.

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Magma

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A004591 Expansion of sqrt(10) in base 8.

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