A066666 -id:A066666 - cp's OEIS frontend

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

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.

A275967 Decimal expansion of the area cut out by a rotating regular pentagon of width 1 inside a unit square.

Original entry on oeis.org

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

Offset: 0

Jeremy Tan, Aug 14 2016

The part of the square not cut out is made of four congruent regions, one at each corner. Superficially they look like isosceles right triangles, but their "hypotenuses" curve very slightly outward. The region cut out by the pentagon is therefore concave.

Jeremy Tan, Regular pentagon in a square (with derivation of constant)

Cf. A066666 (rotating Reuleaux triangle in a square).

First@ RealDigits@ N[1 - (Sqrt@ 5 - 2)/10 Pi, 120] (* Michael De Vlieger, Aug 15 2016 *)

PARI
```
1-(sqrt(5)-2)*Pi/10
```

A = 1-(sqrt(5)-2)*Pi/10 = 0.92583705761388600837689121233541...

A377935 Decimal expansion of the area enclosed by the locus of the centroid of a Reuleaux triangle.

Original entry on oeis.org

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

Offset: 0

Kritsada Moomuang, Nov 11 2024

			0.0793295472807257146962860633...

Eric Weisstein's World of Mathematics, Reuleaux Triangle.

Cf. A066666.

RealDigits[4 - 8/Sqrt[3] + (2*(Pi/9)), 10, 100, -1][[1]]

Equals 4 - 8/sqrt(3) + 2*Pi/9.

Equals (4/9)*(Pi - 3*A066666).

cp's OEIS Frontend

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

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A275967 Decimal expansion of the area cut out by a rotating regular pentagon of width 1 inside a unit square.

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A377935 Decimal expansion of the area enclosed by the locus of the centroid of a Reuleaux triangle.

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