A060708 -id:A060708 - cp's OEIS frontend

A060709 Continued fraction expansion of the Reuleaux Triangle constant.

0, 1, 2, 2, 1, 1, 2, 1, 1, 9, 2, 10, 3, 1, 7, 1, 13, 1, 26, 1, 1, 2, 12, 2, 4, 3, 11, 1, 1, 11, 1, 4, 1, 1, 10, 1, 1, 2, 793, 2, 1, 1, 5, 1, 3, 2, 3, 2, 3, 3, 1, 3, 29, 2, 1, 3, 1, 2, 2, 4, 13, 1, 7, 2, 112, 1, 1, 1, 2, 3, 1, 5, 1, 1, 4

Offset: 1

Robert G. Wilson v, Feb 05 2001

			0.704770923010457972467598520... = 0 + 1/(1 + 1/(2 + 1/(2 + 1/(1 + ...)))). - _Harry J. Smith_, Jul 09 2009

Martin Gardner, "The Unexpected Hanging," page 215.

Harry J. Smith, Table of n, a(n) for n = 1..20000

Cf. A060708.

ContinuedFraction[ (Pi - Sqrt[3])/2, 75]

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac((Pi - sqrt(3))/2); for (n=1, 20000, write("b060709.txt", n, " ", x[n])); } \\ Harry J. Smith, Jul 09 2009

A066666 Decimal expansion of area cut out by a rotating Reuleaux triangle.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Jan 11 2002

"Yes - there are shapes of constant width other than the circle. No - you can't drill square holes. But saying this was not just an attention catcher. As the applet on the right illustrates, you can drill holes that are almost square - drilled holes whose border includes straight line segments!" - Bogomolny. The Java applet shows it in its three versions.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.3.1, p. 490.
Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009.

Anonymous, Shape traced out by a rotating Reuleaux drill.
Alexander Bogomolny, Shapes of constant width.
Eric Weisstein's World of Mathematics, Reuleaux Triangle.

Cf. A060708, A060709.

RealDigits[N[2*Sqrt[3] + Pi/6 - 3, 100]]

2*sqrt(3) + Pi/6 - 3 \\ Stefano Spezia, Dec 21 2024

Area = 2*Sqrt(3)+Pi/6 - 3 = 0.9877003907360534601319999...

A216606 Decimal expansion of 360/7.

Original entry on oeis.org

5, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7

Offset: 2

Paul Curtz, Sep 10 2012

A020806 preceded by a 5.

Number of degrees in the exterior angle of an equilateral heptagon. Since 1969, used in many (orbiform or Reuleaux) heptagonal coins. Zambia has a natural heptagonal coin. Brazil and Costa Rica have a coin with the natural heptagon inscribed in the coin's disk.

			51.42857...

World of Coins, An Alphabet of Heptagons: Seven-sided Coins.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A019695, A020806, A021018, A060708, A068028, A178817, A178818.

Digits:=100: evalf(360/7); # Wesley Ivan Hurt, Jun 28 2016

Flatten[RealDigits[360/7, 10, 100]] (* Wesley Ivan Hurt, Jun 28 2016 *)

360/7. \\ Charles R Greathouse IV, Sep 12 2012

a(n) = 50 + 10*A020806(n).
After 5, of period 6: repeat [1, 4, 2, 8, 5, 7].
From Wesley Ivan Hurt, Jun 28 2016: (Start)
G.f.: x^3*(5-4*x+3*x^2+3*x^3+2*x^4) / (1-x+x^3-x^4).
a(n) = 9/2 + 11*cos(n*Pi)/6 + 5*cos(n*Pi/3)/3 + sqrt(3)*sin(n*Pi/3), n>2.
a(n) = a(n-1) - a(n-3) + a(n-4) for n>6, a(n) = a(n-6) for n>8. (End)

A244046 Decimal expansion of the maximal width of a Reuleaux triangle avoiding all vertices of the integer square lattice.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 18 2014

			1.54494170037159315418509683847062658...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.10 Reuleaux triangle constants, p. 515.

Eric Weisstein's MathWorld, Reuleaux Triangle
Wikipedia, Reuleaux Triangle

Cf. A060708, A066666, A137615.

w = Root[4*x^6 - 12*x^5 + x^4 + 22*x^3 - 14*x^2 - 4*x + 4, x, 3]; RealDigits[w, 10, 105] // First

Smallest positive root of 4*x^6 - 12*x^5 + x^4 + 22*x^3 - 14*x^2 - 4*x + 4.

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A060709 Continued fraction expansion of the Reuleaux Triangle constant.

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A066666 Decimal expansion of area cut out by a rotating Reuleaux triangle.

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A216606 Decimal expansion of 360/7.

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A244046 Decimal expansion of the maximal width of a Reuleaux triangle avoiding all vertices of the integer square lattice.

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