Implementing an unusual sequence as an infinite list in Haskell

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I have two elements as the beginning of the list [1, 2]

This unusual sequence is that it replicates the digits in the number of digits of a certain type that follows the three elements. For example, after 1 and 2, we would have another 2, followed by two 1's. The first few elements of the desired list would yield

[1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2] because

1  2   2  1 1   2  1  2   2  1   2   2

where the previous digits represent the length of the mini-sequences of the same digit.

So far I've tried using the replicate function for repeating the same digit based on an element earlier in the list.

selfrle :: [Int]
selfrle = 1 : 2 : [x | a <- [0..], let x = replicate (selfrle !! a) (selfrle !! (a + 1))) ]

The problem is I can't figure out why it's not working.


Not that unusual as it appears in the OEIS at https://oeis.org/A000002 and is named there as the Kolakoski sequence. There they even give a Haskell program by John Tromp dated 2011:

a = 1:2: drop 2 (concat . zipWith replicate a . cycle $ [1, 2])

Infinite list tricks in Haskell, In this article we use simple sequences as lists of infinite length in a Haskell gives two ways of implementing this sequence as an infinite list  Another common example when demonstrating infinite lists is the Fibonacci sequence -- Wikipedia's page on Haskell gives two ways of implementing this sequence as an infinite list -- I'll add


The first thing you can do it get your types right. replicate builds a list, so the x in your list comprehension has the type [Int], and the entire list comprehension is a list of all the x values, and its type is [[Int]]. You cannot use a list of lists of Int as the tail of a list, when the first two elements are 1 and 2. The types just don't match: you need to decide whether this is a list of Int, or a list of lists of Int.

Based on your description, I suspect you want to fix this by flattening the list comprehension using concat, like so:

selfrie = 1 : 2 : concat [x | a <- [0..],
                              let x = replicate (selfrie !! a) (selfrie !! (a + 1))]

If you try this, you won't get the list you described. I don't know how to help with the next part, and that's because I don't understand your description of the desired list. This isn't a programming question so much as a specification question, so perhaps you could go back to the original source and see how it's explained there?

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We can write this slightly more elegant than @DanD's answer, by using an extra variable b on which we "tie a knot":

a :: [Int]
a = 1 : 2 : b
    where b = 2 : concat (zipWith replicate b (cycle [1, 2]))

We here thus first yield 1 : 2 : b, with b that starts with 2 and then the concatenation of the replicates of b with an endless list with [1, 2, 1, 2, …].

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