My favorite near miss not documented in the article comes from the golden ratio 1.618033988749895…
Repeated powers of the golden ratio are remarkably close to integers. By the time we raise the golden ratio to about the 72nd power, IEEE 754 double-precision floats cannot tell that it is not an integer.
On one hand, every power of the golden ratio is irrational. However, on the other hand, the golden ratio's powers encode the Fibonacci sequence as a recurrence relation, so that the powers tend to be integers. This tension is what creates the near misses.
You can basically make your own kind of 'near misses' with recurrent relations. If you have a recurrent relation of the type f(n + 1) = a * f(n) + b * f(n - 1), you (almost always) get a solution of the form f(n) = c * alpha^n + d * beta^n, where alpha and beta depend only on a and b (specifically, they are the root of the characteristic polynomial, see [1]), and c and d can be determined by the seed values of f. By choosing integers a and b such that the absolute value of beta is less than one (but not zero), you will have that c * alpha^n approximates f(n), since d * beta^n will tend to zero for large n. At the same time f(n) will produce integer values, since we picked integers for a and b.
Repeated powers of the golden ratio are remarkably close to integers. By the time we raise the golden ratio to about the 72nd power, IEEE 754 double-precision floats cannot tell that it is not an integer.
On one hand, every power of the golden ratio is irrational. However, on the other hand, the golden ratio's powers encode the Fibonacci sequence as a recurrence relation, so that the powers tend to be integers. This tension is what creates the near misses.