(3,−2) || (4,−2) || style="color:black" | (5,−2) |- | style="color:black" | (−5,−3) || style="color:#bc1e47" | (−4,−3) || (−3,−3) || (−2,−3) || (−1,−3) || (0,−3) || (1,−3) || (2,−3) || (3,−3) || style="color:#bc1e47" | (4,−3) || style="color:black" | (5,−3) |- | style="color:black" | (−5,−4) || style="color:black" | (−4,−4) || style="color:#bc1e47" | (−3,−4) || (−2,−4) || (−1,−4) || (0,−4) || (1,−4) || (2,−4) || style="color:#bc1e47" | (3,−4) || style="color:black" | (4,−4) || style="color:black" | (5,−4) |- style="color:black" | (−5,−5) || (−4,−5) || (−3,−5) || (−2,−5) || (−1,−5) || style="color:#bc1e47" | (0,−5) || (1,−5) || (2,−5) || (3,−5) || (4,−5) || (5,−5) |} The 12 cells (0, ±5), (±5, 0), (±3, ±4), (±4, ±3) are exactly on the circle, and 69 cells are completely inside, so the approximate area is 81, and is calculated to be approximately 3.24 because = 3.24. Results for some values of are shown in the table below: For related results see The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n. Similarly, the more complex approximations of given below involve repeated calculations of some sort, yielding closer and closer approximations with increasing numbers of calculations. Continued fractions Besides its simple continued fraction representation [3; 7, 15, 1, 292, 1, 1,...], which displays no discernible pattern, has many generalized continued fraction representations generated by a simple rule, including these two. The well-known values and are respectively the second and fourth continued fraction approximations to π. (Other representations are available at The Wolfram Functions Site.) Trigonometry Gregory–Leibniz series The Gregory–Leibniz series is the power series for arctan(x) specialized to = 1. It converges too slowly to be of practical interest. However, the power series converges much faster for smaller values of , which leads to formulae where arises as the sum of small angles with rational tangents, known as Machin-like formulae. Arctangent Knowing that 4 arctan 1 = , the formula can be simplified to get: with a convergence such that each additional 10 terms yields at least three more digits. Another formula for involving arctangent function is given by where such that . Approximations can be made by using, for example, the rapidly convergent Euler formula Alternatively, the following simple expansion series of the arctangent function can be used where to approximate with even more rapid convergence. Convergence in this arctangent formula for improves as integer increases. The constant can also be expressed