A394451 Decimal expansion of the area of a 20-gon with unit side lengths.

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

Offset

2,1

Comments

The largest solution to x^4 - 20*x^3 - 350*x^2 - 500*x + 625 = 0.

This constant multiplied by the square of the side length of a regular 20-gon equals the area of that 20-gon.

20^2 divided by this constant equals 80 * tan(Pi/20) = 12.67075522... which is the perimeter and the area of an equable 20-gon with its side length 4 * tan(Pi/20) = 0.63353776... .

Links

Table of n, a(n) for n=2..106.

Wikipedia, Icosagon.

Index entries for algebraic numbers, degree 4.

Formula

Equals 5 / tan(Pi/20).

Equals 5 / (1 + sqrt(5) - sqrt(5 + sqrt(20))).

Equals 5 * tan(9*Pi/20).

Equals 5 * (1 + sqrt(5) + sqrt(5 + sqrt(20))).

Equals 5 + 10 * (sqrt(5/4) + sqrt(5/4 + sqrt(5/4))) = 5 + 10 * A188594.

Equals 5 * phi * (2 + sqrt(5/2 + sqrt(5/4))) where phi = 1.61803398... .

Example

31.5687575733752154...

Maple

evalf((5 / tan(Pi/20)), 105);

Mathematica

RealDigits[5 / Tan[Pi/20], 10, 105][[1]]

Prog

(PARI) 5 / tan(Pi/20)

Crossrefs

Cf. A001622, A019907, A019979, A188594.

Sequence in context: A140950 A256504 A205713 * A254344 A278032 A016600

Adjacent sequences: A394448 A394449 A394450 * A394452 A394453 A394454

Keyword

nonn,cons

Author

Michal Paulovic, Mar 20 2026

Status

approved