Performs an Engel expansion of the given rational number into a sum of fractions of the form 1/a + 1/(a*b) + 1/(a*b*c)....

This function returns only the denominators of the expansion (i.e., only [a, a*b, ...]). If the given rational number r satisfies r ≥ 1, the first n elements of the expansion will be 1, where n = floor(r). If the given rational number r satisfies r < 0, all returned denominators will be less than 0, so that the relationship

r == sum(1 .// efengel(r))

is always true.

Performs a greedy (Fibonacci–Sylvester) expansion of the given rational number into a sum of fractions of the form 1/a_1 + 1/a_2 + ...

This function returns only the denominators of the expansion (i.e., only [a_1, a_2, ...]). If the given rational number r satisfies r ≥ 1, the first n elements of the expansion will be 1, where n = floor(r). If the given rational number r satisfies r < 0, all returned denominators will be less than 0, so that the relationship

r == sum(1 .// efgreedy(r))

is always true.

Performs a harmonic expansion of the given rational number into a sum of fractions of the form 1/2 + 1/3 + ..., and concludes the remainder of the expansion using efgreedy if the given rational number cannot be represented as a sum in the harmonic sequence.

If the second argument is specified, the expansion will begin using that value as the first denominator. In other words, while efharmonic(r) will return an array beginning [2, 3, ...], efharmonic(r, 3) will return an array beginning [3, 4, ...], and so on.

This function will throw an exception if the second argument f ≤ 1.

This function returns only the denominators of the expansion (i.e., only [a_1, a_2, ...]). If the given rational number r satisfies r ≥ 1, the first n elements of the expansion will be 1, where n = floor(r). If the given rational number r satisfies r < 0, all returned denominators will be less than 0, so that the relationship

r == sum(1 .// efharmonic(r))

is always true.

Performs a greedy (Fibonacci–Sylvester) expansion of the given rational number into a sum of fractions of the form 1/a_1 + 1/a_2 + ..., but only using odd denominators.

It can be shown that, for any rational number x/y where y is odd, you can write this as a finite sum of fractions where all denominators are odd. This method sometimes produces an expansion with fewer elements than the traditional greedy algorithm.

This function will throw an exception if the denominator of the rational number supplied is even.

This function returns only the denominators of the expansion (i.e., only [a_1, a_2, ...]). If the given rational number r satisfies r ≥ 1, the first n elements of the expansion will be 1, where n = floor(r). If the given rational number r satisfies r < 0, all returned denominators will be less than 0, so that the relationship

r == sum(1 .// efoddgreedy(r))

is always true.

Performs an Engel expansion of the given rational number into a sum of fractions of the form 1/a + 1/(a*b) + 1/(a*b*c)....

This function returns only the unique denominators of the expansion (i.e., only [a, b, ...]). If the given rational number r satisfies r ≥ 1, the first n elements of the expansion will be 1, where n = floor(r).