A020817 -id:A020817 - cp's OEIS frontend

A010513 Decimal expansion of square root of 60.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 7 followed by {1, 2, 1, 14} repeated. - Harry J. Smith, Jun 07 2009

With a different offset, decimal expansion of 0.6. In a unimodal distribution, the mean and median differ by at most 0.6 standard deviations (and this is sharp), see Basu & DasGupta. - Charles R Greathouse IV, Oct 01 2024

			7.745966692414833770358530799564799221665843410583181653175147532226966....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Sanjib Basu and Anirban DasGupta, The Mean, Median, and Mode of Unimodal Distributions: A Characterization, Theory of Probability & Its Applications 41:2 (1997), pp. 210-223; alternative link.
Index entries for algebraic numbers, degree 2.

Cf. A040052 (continued fraction).

Cf. A010472, A011053, A020772, A020817.

RealDigits[N[Sqrt[60],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 25 2011 *)

{ default(realprecision, 20080); x=sqrt(60); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010513.txt", n, " ", d)); } \\ Harry J. Smith, Jun 07 2009

Equals 10 * sqrt(3/5) = 10 * Sum_{k>=0} (-1)^k * binomial(2*k,k)/6^k. - Amiram Eldar, Aug 03 2020

Equals 2*A010472 = A011053^2 = 30*A020772 = 1/A020817. - Hugo Pfoertner, Oct 02 2024

A041105 Denominators of continued fraction convergents to sqrt(60).

Original entry on oeis.org

1, 1, 3, 4, 59, 63, 185, 248, 3657, 3905, 11467, 15372, 226675, 242047, 710769, 952816, 14050193, 15003009, 44056211, 59059220, 870885291, 929944511, 2730774313, 3660718824, 53980837849, 57641556673, 169263951195, 226905507868, 3345941061347, 3572846569215

Offset: 0

N. J. A. Sloane

Interspersion of 4 linear recurrences with constant coefficients. - Gerry Martens, Jun 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,62,0,0,0,-1).

Cf. A041104, A040052, A020817, A010513.

Cf. A258684.

I:=[1, 1, 3, 4, 59, 63, 185, 248]; [n le 8 select I[n] else 62*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

numtheory:-cfrac(sqrt(60),100,'con','den'):
den[1..-2]; # Robert Israel, Jun 09 2015

Denominator[Convergents[Sqrt[60], 30]] (* Vincenzo Librandi, Dec 11 2013 *)
d0 := LinearRecurrence[{62, -1}, {1, 59}, 20]
d1 := LinearRecurrence[{62, -1}, {1, 63}, 20] (* A258684  *)
d2 := LinearRecurrence[{62, -1}, {3, 185}, 20]
d3 := LinearRecurrence[{62, -1}, {4, 248}, 20]
Flatten[MapIndexed[{d0[[#]] , d1[[#]], d2[[#]] , d3[[#]]} &,
  Range[10]]] (* Gerry Martens, Jun 09 2015 *)
LinearRecurrence[{0, 0, 0, 62, 0, 0, 0, -1},{1, 1, 3, 4, 59, 63, 185, 248},30] (* Ray Chandler, Aug 03 2015 *)

G.f.: -(x^2-x-1)*(x^4+4*x^2+1) / ((x^4-8*x^2+1)*(x^4+8*x^2+1)). - Colin Barker, Nov 12 2013

a(n) = 62*a(n-4) - a(n-8). - Vincenzo Librandi, Dec 11 2013

More terms from Colin Barker, Nov 12 2013

A176020 Decimal expansion of (3+sqrt(15))/3.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 06 2010

Continued fraction expansion of (3+sqrt(15))/3 is A010693.

a(n) = A020817(n-1) for n > 1; a(1) = 2.

			(3+sqrt(15))/3 = 2.29099444873580562839...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010472 (decimal expansion of sqrt(15)), A176016 (decimal expansion of (3+sqrt(15))/6), A010693 (repeat 2, 3), A020817 (decimal expansion of 1/sqrt(60)).

RealDigits[(3+Sqrt[15])/3,10,120][[1]] (* Harvey P. Dale, May 20 2013 *)

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A010513 Decimal expansion of square root of 60.

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A041105 Denominators of continued fraction convergents to sqrt(60).

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A176020 Decimal expansion of (3+sqrt(15))/3.

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