A002193 -id:A002193 - cp's OEIS frontend

A156163 Decimal expansion of (19+6*sqrt(2))/17.

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

Offset: 1

Klaus Brockhaus, Feb 09 2009

lim_{n -> infinity} b(n)/b(n-1) = (19+6*sqrt(2))/17 for n mod 3 = {0, 2}, b = A155923.

lim_{n -> infinity} b(n)/b(n-1) = ((19+6*sqrt(2))/17)^2 for n mod 3 = {0, 2}, b = A156159.

			(19+6*sqrt(2))/17 = 1.61678125730815119369...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2

Cf. A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)).

RealDigits[(19 + 6*Sqrt[2])/17, 10, 100][[1]] (* G. C. Greubel, Jul 05 2017 *)

(6*sqrt(2)+19)/17 \\ Charles R Greathouse IV, Apr 25 2016

A157697 Decimal expansion of sqrt(2/3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 04 2009

Height (from a vertex to the opposite face) of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
The eccentricity of the ellipse of minimum area that is circumscribing two equal and externally tangent circles (Kotani, 1995). - Amiram Eldar, Mar 06 2022
The standard deviation of a roll of a 3-sided die. - Mohammed Yaseen, Feb 23 2023

			0.81649658092772603273242802490196379732198249355222...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (168) on page 32.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Jisho Kotani, Problem 2053, Crux Mathematicorum, Vol. 5, No. 1 (1995), p. 202; Solution to Problem 2053, by David Hankin, ibid., Vol. 22, No. 4 (1996), pp. 187-188.
D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly, Vol. 92, No. 7 (1985), pp. 449-457.
Index entries for algebraic numbers, degree 2.

Cf. A002193, A020763, A092259, A115754, A145439.

Sqrt(2/3); // G. C. Greubel, Mar 30 2018

Maple
```
evalf(sqrt(2/3)) ;
```

RealDigits[Sqrt[2/3], 10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

sqrt(2/3) \\ G. C. Greubel, Mar 30 2018

Equals 1 - (1/2)/2 + (1*3)/(2*4)/2^2 - (1*3*5)/(2*4*6)/2^3 + ... [Jolley]
Equals Sum_{n>=0} (-1)^n*binomial(2n,n)/8^n = 1/A115754. Averaging this constant with sqrt(2) = A002193 = Sum_{n>=0} binomial(2n,n)/8^n yields A145439.
From Michal Paulovic, Dec 08 2022: (Start)
Equals 2 * A020763.
Has periodic continued fraction expansion [0, 1, 4; 2, 4]. (End)
Equals exp(-arctanh(1/5)). - Amiram Eldar, Jul 10 2023
Equals Product_{k>=1} (1 + (-1)^k/A092259(k)). - Amiram Eldar, Nov 24 2024

A322641 Number of times the digit 0 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

Original entry on oeis.org

0, 10, 108, 952, 9959, 99814, 999897, 10002237, 100010228, 999996989

Offset: 1

Martin Renner, Dec 21 2018

It is not known if sqrt(2) is normal, but the distribution of decimal digits found for the first 10^n digits of sqrt(2) shows no statistically significant departure from a uniform distribution.

Eric Weisstein's World of Mathematics, Pythagoras's Constant Digits.

Cf. A002193, A099291, A322642, A322643, A322644, A322645, A322646, A322647, A322648, A322649, A322650.

a:=proc(n)
  local digits, SQRT2, C, i;
  digits:=10^n+100;
  SQRT2:=convert(frac(evalf[digits](sqrt(2))),string)[2..digits-99];
  C:=0;
  for i from 1 to length(SQRT2) do
    if SQRT2[i]="0" then C:=C+1; fi;
  od;
  return(C);
end;

Table[DigitCount[IntegerPart[(Sqrt[2]-1)*10^10^n], 10, 0], {n,1,10}] (* Robert Price, Mar 29 2019 *)

A322642 Number of times the digit 1 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

Original entry on oeis.org

2, 7, 98, 1005, 10106, 98924, 1000114, 10000179, 99998381, 1000042849

Offset: 1

Martin Renner, Dec 21 2018

It is not known if sqrt(2) is normal, but the distribution of decimal digits found for the first 10^n digits of sqrt(2) shows no statistically significant departure from a uniform distribution.

Eric Weisstein's World of Mathematics, Pythagoras's Constant Digits.

Cf. A002193, A099292, A322641, A322643, A322644, A322645, A322646, A322647, A322648, A322649, A322650.

a:=proc(n)
  local digits, SQRT2, C, i;
  digits:=10^n+100;
  SQRT2:=convert(frac(evalf[digits](sqrt(2))),string)[2..digits-99];
  C:=0;
  for i from 1 to length(SQRT2) do
    if SQRT2[i]="1" then C:=C+1; fi;
  od;
  return(C);
end;

Table[DigitCount[IntegerPart[(Sqrt[2]-1)*10^10^n], 10, 1], {n,1,10}] (* Robert Price, Mar 29 2019 *)

A322643 Number of times the digit 2 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

Original entry on oeis.org

2, 8, 109, 1004, 9876, 100436, 1000208, 9998091, 99995645, 999987069

Offset: 1

Martin Renner, Dec 21 2018

It is not known if sqrt(2) is normal, but the distribution of decimal digits found for the first 10^n digits of sqrt(2) shows no statistically significant departure from a uniform distribution.

Eric Weisstein's World of Mathematics, Pythagoras's Constant Digits.

Cf. A002193, A099293, A322641, A322642, A322644, A322645, A322646, A322647, A322648, A322649, A322650.

a:=proc(n)
  local digits, SQRT2, C, i;
  digits:=10^n+100;
  SQRT2:=convert(frac(evalf[digits](sqrt(2))),string)[2..digits-99];
  C:=0;
  for i from 1 to length(SQRT2) do
    if SQRT2[i]="2" then C:=C+1; fi;
  od;
  return(C);
end;

Table[DigitCount[IntegerPart[(Sqrt[2]-1)*10^10^n], 10, 2], {n,1,10}] (* Robert Price, Mar 29 2019 *)

A322644 Number of times the digit 3 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

Original entry on oeis.org

2, 11, 82, 980, 10058, 100191, 999674, 10004178, 99995415, 999984900

Offset: 1

Martin Renner, Dec 21 2018

It is not known if sqrt(2) is normal, but the distribution of decimal digits found for the first 10^n digits of sqrt(2) shows no statistically significant departure from a uniform distribution.

Eric Weisstein's World of Mathematics, Pythagoras's Constant Digits.

Cf. A002193, A099294, A322641, A322642, A322643, A322645, A322646, A322647, A322648, A322649, A322650.

a:=proc(n)
  local digits, SQRT2, C, i;
  digits:=10^n+100;
  SQRT2:=convert(frac(evalf[digits](sqrt(2))),string)[2..digits-99];
  C:=0;
  for i from 1 to length(SQRT2) do
    if SQRT2[i]="3" then C:=C+1; fi;
  od;
  return(C);
end;

Table[DigitCount[IntegerPart[(Sqrt[2]-1)*10^10^n], 10, 3], {n,1,10}] (* Robert Price, Mar 29 2019 *)

A322645 Number of times the digit 4 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

Original entry on oeis.org

2, 9, 100, 1016, 10100, 100024, 1000126, 10000054, 100012725, 1000008724

Offset: 1

Martin Renner, Dec 21 2018

It is not known if sqrt(2) is normal, but the distribution of decimal digits found for the first 10^n digits of sqrt(2) shows no statistically significant departure from a uniform distribution.

Eric Weisstein's World of Mathematics, Pythagoras's Constant Digits.

Cf. A002193, A099294, A322641, A322642, A322643, A322644, A322646, A322647, A322648. A322649, A322650.

a:=proc(n)
  local digits, SQRT2, C, i;
  digits:=10^n+100;
  SQRT2:=convert(frac(evalf[digits](sqrt(2))),string)[2..digits-99];
  C:=0;
  for i from 1 to length(SQRT2) do
    if SQRT2[i]="4" then C:=C+1; fi;
  od;
  return(C);
end;

Table[DigitCount[IntegerPart[(Sqrt[2]-1)*10^10^n], 10, 4], {n,1,10}] (* Robert Price, Mar 29 2019 *)

A322646 Number of times the digit 5 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

Original entry on oeis.org

1, 7, 104, 1001, 10002, 100155, 999358, 9998344, 100002636, 999970045

Offset: 1

Martin Renner, Dec 21 2018

It is not known if sqrt(2) is normal, but the distribution of decimal digits found for the first 10^n digits of sqrt(2) shows no statistically significant departure from a uniform distribution.

Eric Weisstein's World of Mathematics, Pythagoras's Constant Digits.

Cf. A002193, A099296, A322641, A322642, A322643, A322644, A322645, A322647, A322648, A322649, A322650.

a:=proc(n)
  local digits, SQRT2, C, i;
  digits:=10^n+100;
  SQRT2:=convert(frac(evalf[digits](sqrt(2))),string)[2..digits-99];
  C:=0;
  for i from 1 to length(SQRT2) do
    if SQRT2[i]="5" then C:=C+1; fi;
  od;
  return(C);
end;

Table[DigitCount[IntegerPart[(Sqrt[2]-1)*10^10^n], 10, 5], {n,1,10}] (* Robert Price, Mar 29 2019 *)

A322647 Number of times the digit 6 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

Original entry on oeis.org

1, 10, 90, 1032, 9939, 99886, 1001246, 10001665, 100012683, 1000007824

Offset: 1

Martin Renner, Dec 21 2018

It is not known if sqrt(2) is normal, but the distribution of decimal digits found for the first 10^n digits of sqrt(2) shows no statistically significant departure from a uniform distribution.

Eric Weisstein's World of Mathematics, Pythagoras's Constant Digits.

Cf. A002193, A099297, A322641, A322642, A322643, A322644, A322645, A322646, A322648, A322649, A322650.

a:=proc(n)
  local digits, SQRT2, C, i;
  digits:=10^n+100;
  SQRT2:=convert(frac(evalf[digits](sqrt(2))),string)[2..digits-99];
  C:=0;
  for i from 1 to length(SQRT2) do
    if SQRT2[i]="6" then C:=C+1; fi;
  od;
  return(C);
end;

Table[DigitCount[IntegerPart[(Sqrt[2]-1)*10^10^n], 10, 6], {n,1,10}] (* Robert Price, Mar 29 2019 *)
Table[Count[RealDigits[Sqrt[2],10,10^n][[1]],6],{n,10}] (* Harvey P. Dale, Oct 02 2022 *)

A322648 Number of times the digit 7 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

Original entry on oeis.org

0, 18, 104, 964, 10008, 100008, 999359, 9998646, 99980315, 999986743

Offset: 1

Martin Renner, Dec 21 2018

It is not known if sqrt(2) is normal, but the distribution of decimal digits found for the first 10^n digits of sqrt(2) shows no statistically significant departure from a uniform distribution.

Eric Weisstein's World of Mathematics, Pythagoras's Constant Digits.

Cf. A002193, A099298, A322641, A322642, A322643, A322644, A322645, A322646, A322647, A322649, A322650.

a:=proc(n)
  local digits, SQRT2, C, i;
  digits:=10^n+100;
  SQRT2:=convert(frac(evalf[digits](sqrt(2))),string)[2..digits-99];
  C:=0;
  for i from 1 to length(SQRT2) do
    if SQRT2[i]="7" then C:=C+1; fi;
  od;
  return(C);
end;

Table[DigitCount[IntegerPart[(Sqrt[2]-1)*10^10^n], 10, 7], {n,1,10}] (* Robert Price, Mar 29 2019 *)

cp's OEIS Frontend

A156163 Decimal expansion of (19+6*sqrt(2))/17.

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A157697 Decimal expansion of sqrt(2/3).

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A322641 Number of times the digit 0 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

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A322642 Number of times the digit 1 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

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A322643 Number of times the digit 2 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

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A322644 Number of times the digit 3 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

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A322645 Number of times the digit 4 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

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A322646 Number of times the digit 5 appears in the first 10^n decimal digits of sqrt(2), sometimes called Pythagoras's constant, counting after the decimal point.

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