A344069 -id:A344069 - cp's OEIS frontend

A010470 Decimal expansion of square root of 13.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 3 followed by {1, 1, 1, 1, 6} repeated. - Harry J. Smith, Jun 02 2009
The convergents to sqrt(13) are given in A041018/A041019. - Wolfdieter Lang, Nov 23 2017
The fundamental algebraic (integer) number in the field Q(sqrt(13)) is (1 + sqrt(13))/2  = A209927. - Wolfdieter Lang, Nov 21 2023

			3.605551275463989293119221267470495946251296573845246212710453056227166...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.31.4, p. 201.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2.

Cf. A010122 (continued fraction), A041018/A041019 (convergents), A248242 (Egyptian fraction), A171983 (Beatty sequence).

Cf. A020770 (reciprocal), A209927, A295330, A344069.

RealDigits[N[Sqrt[13],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=sqrt(13); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010470.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

A344382 Decimal expansion of sqrt(29)/5.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 18 2021

Essentially the same as A188730 after the first two initial terms.

sqrt(29)/5 is the length of the shortest line segment needed to dissect the unit square into 5 regions with equal areas if all the line segments start at the same vertex of the square.

			1.07703296142690080625014209...

Cf. A344069, A188730.

Mathematica
```
RealDigits[Sqrt[29]/5, 10, 200][[1]]
```

A381485 Decimal expansion of sqrt(13)/6.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Feb 24 2025

The greatest possible minimum distance between 6 points in a unit square.

The solution was found by Ronald L. Graham and reported by Schaer (1965).

			0.60092521257733154885320354457841599104188276230754...

Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.
D. Würtz, M. Monagan, and R. Peikert, The history of packing circles in a square, Maple Technical Newsletter, Vol. 1 (1994), pp. 35-42; ResearchGate link.
Index entries for algebraic numbers, degree 2.

Solutions for k points: A002193 (k = 2), A120683 (k = 3), 1 (k = 4), A010503 (k = 5), this constant (k = 6), A379338 (k = 7), A101263 (k = 8), A020761 (k = 9), A281065 (k = 10).

Cf. A010470, A142464, A176019, A295330, A344069.

Mathematica
```
RealDigits[Sqrt[13] / 6, 10, 120][[1]]
```

list(len) = digits(floor(10^len*quadgen(52)/6));

Equals A010470 / 6 = A295330 / 3 = A344069 / 2 = A176019 - 1/2 = sqrt(A142464).

Minimal polynomial: 36*x^2 - 13.

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A010470 Decimal expansion of square root of 13.

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A344382 Decimal expansion of sqrt(29)/5.

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A381485 Decimal expansion of sqrt(13)/6.

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