The ±0.8 correction
A square of side n·√0.2 has area n²/5, which is almost never a whole number. The Pentametric Formula moves that area by exactly +0.8, −0.8 or 0 to reach the integer P(n), and the matching circles move by exactly ±2π/5. Which way it moves is decided by the last digit of n.
P(n) − n²/5 = ⁴⁄₅ · n5 ∈ {0, +0.8, −0.8}
n =
The circle through the corners of a square of area A has area πA/2, so a shift of 4/5 in the square becomes a shift of 2π/5 (a fifth of a turn) in the circle.
The correction for n = 0 … 50
Click a bar to inspect that n. At every Lucas number the correction is never 0, and its sign alternates: −0.8 at Lk with k even, +0.8 with k odd, which is why P(Lk) = Fk2 exactly.
The last-digit rule
The sign comes from the last digit of the double square 2n² (2, 8, 18, 32, 50, …): it ends in 2 for +0.8, in 8 for −0.8 and in 0 for no correction. This is the Legendre symbol n5 in disguise.