A010470 -id:A010470 - cp's OEIS frontend

A194393 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(13) and < > denotes fractional part.

2, 4, 6, 8, 24, 26, 28, 30, 32, 38, 40, 42, 44, 46, 62, 64, 66, 68, 70, 76, 78, 80, 82, 84, 100, 102, 104, 106, 108, 110, 112, 138, 140, 142, 144, 146, 148, 150, 176, 178, 180, 182, 184, 186, 188, 204, 206, 208, 210, 212, 218, 220, 222, 224, 226, 242, 244

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Cf. A010470, A194368, A194392, A194394.

r = Sqrt[13]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t1, 1]]       (* A194392 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]       (* A194393 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]       (* A194394 *)

A194394 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(13) and < > denotes fractional part.

Original entry on oeis.org

5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 27, 43, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 65, 81, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 103, 109, 111, 113, 114, 115, 116, 117

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Cf. A010470, A194368, A194392, A194393.

r = Sqrt[13]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t1, 1]]       (* A194392 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]       (* A194393 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]       (* A194394 *)

A177157 Decimal expansion of sqrt(221).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, May 03 2010

Continued fraction expansion of sqrt(221) is A040206.

			sqrt(221) = 14.86606874731850552261...

Cf. A010470 (decimal expansion of sqrt(13)), A010473 (decimal expansion of sqrt(17)), A177156 (decimal expansion of (9+sqrt(221))/14), A040206 (14 followed by (repeat 1, 6, 2, 6, 1, 28)).

A344069 Decimal expansion of sqrt(13)/3.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 08 2021

Draw two congruent line segments inside the unit square from the same vertex of the square to opposite sides of the square. sqrt(13)/3 is the length of each line segment needed to dissect the square into 3 regions with equal areas.

			1.201850425154663097706...

Cf. A010470.

Mathematica
```
RealDigits[Sqrt[13]/3, 10, 100][[1]]
```

A020770 Decimal expansion of 1/sqrt(13).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

1/sqrt(13) = 0.277350098112614561009170866728499688173176659526557400977727158171320534484... [Vladimir Joseph Stephan Orlovsky, May 30 2010]

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A010470.

RealDigits[N[1/Sqrt[13],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

A172274 a(n) = floor(n*(sqrt(13)-sqrt(11))).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23

Offset: 0

Vincenzo Librandi, Jan 30 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A010470, A010468, A172264, A172266-A172270, A172272-A172278.

[Floor(n*(Sqrt(13)-Sqrt(11))): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

With[{c = Sqrt[13] - Sqrt[11]}, Floor[c Range[0, 80]]] (* Vincenzo Librandi, Aug 01 2013 *)

a(n) = floor(n*(A010470 - A010468)). - Rick L. Shepherd, Jun 17 2010

Edited by Rick L. Shepherd, Jun 17 2010

A177016 Decimal expansion of sqrt(16926).

Original entry on oeis.org

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

Offset: 3

Klaus Brockhaus, May 01 2010

Continued fraction expansion of sqrt(16926) is 130 followed by (repeat 10, 260).

sqrt(16926) = sqrt(2)*sqrt(3)*sqrt(7)*sqrt(13)*sqrt(31).

			sqrt(16926) = 130.09996156801891999550...

Cf. A002193 (decimal expansion of sqrt(2)), A002194 (decimal expansion of sqrt(3)), A010465 (decimal expansion of sqrt(7)), A010470 (decimal expansion of sqrt(13)), A010486 (decimal expansion of sqrt(31)), A177015 (decimal expansion of (124+sqrt(16926))/25).

A177925 Decimal expansion of sqrt(2730).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, May 15 2010

Continued fraction expansion of sqrt(2730) is 52 followed by (repeat 4, 104).

sqrt(2730) = sqrt(2)*sqrt(3)*sqrt(5)*sqrt(7)*sqrt(13).

			sqrt(2730) = 52.24940191045252529379...

Cf. A002193 (decimal expansion of sqrt(2)), A002194 (decimal expansion of sqrt(3)), A002163 (decimal expansion of sqrt(5)), A010465 (decimal expansion of sqrt(7)), A010470 (decimal expansion of sqrt(13)), A177924 (decimal expansion of (28+sqrt(2730))/56).

RealDigits[Sqrt[2730],10,120][[1]] (* Harvey P. Dale, Aug 11 2021 *)

A379533 Decimal expansion of (sqrt(13) - 1)/36.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 24 2024

Lower bound to the 8th Heilbronn triangle constant.

			0.072376424318444147031089479651958220729202682606812...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.16, p. 527.

Francesc Comellas and J. Luis A. Yebra, New lower bounds for Heilbronn numbers, Elec. J. Combin. 9 (2002) R6. See pp. 4-5, 8.

Cf. A010470, A248866, A343851, A379534.

Mathematica
```
RealDigits[(Sqrt[13]-1)/36,10,100][[1]]
```

Minimal polynomial: 108*x^2 + 6*x - 1. - Stefano Spezia, Aug 03 2025

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.

cp's OEIS Frontend

A194393 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(13) and < > denotes fractional part.

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A194394 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(13) and < > denotes fractional part.

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Mathematica

A177157 Decimal expansion of sqrt(221).

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A344069 Decimal expansion of sqrt(13)/3.

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Mathematica

A020770 Decimal expansion of 1/sqrt(13).

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A172274 a(n) = floor(n*(sqrt(13)-sqrt(11))).

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Magma

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Formula

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A177016 Decimal expansion of sqrt(16926).

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A177925 Decimal expansion of sqrt(2730).

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Mathematica

A379533 Decimal expansion of (sqrt(13) - 1)/36.

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