A178816 Decimal expansion of the area of the regular 10-gon (decagon) of edge length 1.

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

Offset

1,1

Comments

An algebraic number with degree 4 and denominator 2; minimal polynomial 16x^4 - 1000x^2 + 3125. - Charles R Greathouse IV, Apr 25 2016

This equals in a regular pentagon inscribed in a unit circle with vertices V0 = (x, y) = (1, 0), and V1..V4 in the counterclockwise sense, one tenth of the y-coordinate of the midpoint of side (V1,V2), named M1: M1_y = (2*sqrt(3 - phi) + sqrt(7 - 4*phi))/4 = sqrt(3 + 4*phi)/4. The x-coordinate is M1_x = -1/4. - Wolfdieter Lang, Jan 09 2018

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10001

Wikipedia, Decagon

Wikipedia, Regular polygon

Index entries for algebraic numbers, degree 4

Formula

Digits of 5*sqrt(5+2*sqrt(5))/2 = (5/2)*sqrt(3 + 4*phi), with phi from A001622.

Example

7.69420884293813350642572644009227456001675535884448106759789062593715...
sqrt(3 + 4*phi)/4 = 0.769420884293813350642572644009227456001675535884... - Wolfdieter Lang, Jan 09 2018

Maple

evalf[120](5*sqrt(5+2*sqrt(5))/2); # Muniru A Asiru, Jan 22 2019

Mathematica

RealDigits[5*Sqrt[5+2*Sqrt[5]]/2, 10, 100][[1]]

Prog

(PARI) 5*sqrt(2*sqrt(5)+5)/2 \\ Charles R Greathouse IV, Apr 25 2016
(Magma) SetDefaultRealField(RealField(100)); 5*Sqrt(2*Sqrt(5)+5)/2; // G. C. Greubel, Jan 22 2019
(Sage) numerical_approx(5*sqrt(2*sqrt(5)+5)/2, digits=100) # G. C. Greubel, Jan 22 2019

Crossrefs

Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A090488, A256853, A256854, A178809.

Cf. A001622.

Sequence in context: A219705 A273066 A257964 * A200106 A201766 A307951

Adjacent sequences: A178813 A178814 A178815 * A178817 A178818 A178819

Keyword

nonn,cons,easy

Author

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

Status

approved