A020772 -id:A020772 - 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

A368088 Index of smallest pentagonal number with n digits.

Original entry on oeis.org

1, 3, 9, 26, 82, 259, 817, 2583, 8166, 25821, 81650, 258200, 816497, 2581990, 8164966, 25819890, 81649659, 258198890, 816496582, 2581988898, 8164965810, 25819888975, 81649658093, 258198889748, 816496580928, 2581988897472, 8164965809278, 25819888974717, 81649658092773

Offset: 1

Kelvin Voskuijl, Dec 17 2023

The digits of the odd- and even-indexed terms converge to those in the decimal expansions of sqrt(2/3) and sqrt(20/3), respectively.

			a(4) = 26 as the 26th pentagonal number is 26*(3*26-1)/2 = 1001 which has 4 digits (while the 25th is 925 which is only 3).

Cf. A000326, A180447.
Cf. A068092 (for triangular numbers), A017936 (for squares).
Cf. A157697 (square root of 2/3), A020772 (square root of 20/3)

a[n_] := Ceiling[(Sqrt[24*10^(n-1) + 1] + 1)/6]; Array[a, 40] (* Amiram Eldar, Dec 30 2023 *)

a(n) = 1 + (sqrtint(24*10^(n-1)) + 1)\6 \\ Andrew Howroyd, Dec 30 2023

a(n) = ceiling((sqrt(24*10^(n-1) + 1) + 1)/6).

cp's OEIS Frontend

A010513 Decimal expansion of square root of 60.

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