Ten forms of one formula
The same areas were reached in the spreadsheets by ten different routes: through digits, floors, a polynomial, cosines, a golden rotation and the angles of the pentagon. Each route is a different way of computing the Legendre symbol n5, so all ten always agree. Choose n and watch them agree.
d = 2n² mod 10 = , r = n² mod 10 = , n5 = , P(n) =
Forms F8 and F9 use cosines and powers of the golden ratio, so they are computed in floating point and agree with P(n) to many decimal places; F9 is evaluated by its recurrence for n ≤ 1400. All the others are exact.
Check them all at once
The paper reports this check for 0 ≤ n < 3000.