A348757 - OEIS

%I #14 Aug 21 2023 11:38:31

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

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

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

%N Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

%C An algebraic number of degree 4. The smaller of the two positive roots of the equation 16*x^4 - 2500*x^2 + 3125 = 0.

%D Robert B. Banks, Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Princeton University Press, 2012, p. 15.

%H Herta T. Freitag, <a href="https://doi.org/10.1111/j.1949-8594.1981.tb15506.x">Problem 3855</a>, School Science and Mathematics, Vol. 81, No. 4 (1981), p. 352; <a href="https://doi.org/10.1111/j.1949-8594.1982.tb17188.x">Solution to Problem 3855</a> by David A. Blaeuer, ibid., Vol. 82, No. 3 (1982), pp. 265-266.

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/753290/area-of-a-five-pointed-star">Area of a five pointed star</a>, 2014.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentagram">Pentagram</a>.

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>

%F Equals 5*sin(Pi/5)/phi^2, where phi is the golden ratio (A001622).

%F Equals 5/(cot(Pi/5) + cot(Pi/10)).

%F Equals 10*tan(Pi/10)/(3 - tan(Pi/10)^2).

%F Equals (5/2)*sqrt((25 -11*sqrt(5))/2).

%F Equals 5*(5 - sqrt(5))/(4*sqrt(5 + 2*sqrt(5))) = A094874 * A179050 = 10 * A094874 / A344172.

%e 1.12256994144896343110486287949381696894803120580270...

%t RealDigits[5*Sin[Pi/5]/GoldenRatio^2, 10, 100][[1]]

%Y Cf. A001622, A019845, A019916, A019952, A019970, A094874, A104955, A134974, A179050, A188593, A344172.

%K nonn,cons

%O 1,3

%A _Amiram Eldar_, Nov 12 2021