Programming Praxis

Aronson’s sequence 1, 4, 11, 16, 24, 29, 33, 35, 39, … (A005224) is an infinite self-referential sequence defined as:

T is the first, fourth, eleventh, … letter in this sentence.

Your task is to write a program that generates Aronson’s sequence and use it to compute the first hundred members of the sequence. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Cardinal numbers are those used for counting; spelled in letters, they are one, two, three, four, and so on.

Ordinal numbers are those used for ranking: spelled in letters, they are first, second, third, fourth, and so on.

Your task is to write a program that takes a number and returns the spelled-out cardinal and ordinal forms of that number. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Lagarias and Sloane study the “approximate squaring” map f(x) = x⌈x⌉ and its behavior when iterated in this paper.

Consider the fraction x = n / d when n > d > 1; let’s take 8/7 as an example. In the first step, the smallest integer greater than 8/7 (the “ceiling”) is 2, and 8/7 × 2 = 16/7. In the second step, the ceiling of 16/7 is 3, so we have 16/7 × 3 = 48/7. And in the third step, the ceiling of 48/7 is 7, so we have 48/7 × 7 = 48. Now the denominator is 1 and the result is an integer, so iteration stops, and we say that 8/7 goes to 48 in 3 steps.

Study shows the iteration is chaotic; sometimes the iteration stops in just a few steps, as in 8/7, sometimes it takes longer (6/5 goes to a number with 57735 digits in 18 steps), and sometimes it’s just ridiculous (200/199 goes to a number with 10⁴³⁵ digits). It is conjectured but not proven that iterated approximate squaring always terminates in an integer.

Your task is to write a program that iterates approximate squaring. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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