There is an infinite chessboard...

Aaah, I did so the summations for the available places...


Starting with the center column, recalling that phi^2 = phi + 1, which means phi^2 - 1 = phi.

1 + 1/phi + 1/phi^2 + 1/phi^3 + ...

= sum(i=0 to inf, 1 / phi^i)
= phi / (phi - 1)
= (phi + 1 - 1) / (phi - 1)
= (phi^2 - 1) / (phi - 1)
= (phi + 1) * (phi - 1) / (phi - 1)
= phi + 1
= phi^2

sum(i=0 to inf, 1 / phi^i) = phi^2

The other columns have the same progression, but divide the total by phi once for every column to the side (and there are 2 sides)

So the sum value of the entire thing is phi^2 (center) + 2 * phi + 2 + 2 / phi + ...

= 2 * phi^2 * sum(i=0 to inf, 1 / phi^i) - phi^2
= 2 * phi^2 * phi^2 - phi^2
= phi^2 * (2 * phi^2 - 1)
= phi^2 * (phi^2 + phi^2 - 1)
= phi^2 * (phi^2 + phi)
= phi^3 * (phi + 1)
= phi^3 * phi^2
= phi^5