Thue

Thue sequences, after Axel Thue, are sequences over an alphabet of three {1,2,3} such that no subsequence is immediately repeated. For example, 1213121 is a Thue sequence that cannot be extended 12131211 x, 12131212 x, 12131213 x. However, there are Thue sequences of any length. (There are no solutions longer than 3 for binary sequences, and finding solutions is easier for alphabets larger than three.)

The 'circular program' below [All93] builds a tree of Thue sequences. Note that within function 'build', the tree data-structure 'T' and functions 'toplevel' and 'f' are mutually recursive. The program relies on the fact that a partial soln 'abcde' can be extended with 'f' iff its shadow, 'bcdef', is already in tree at the previous level; this avoids repeating many tests on constraints that have already been treated at a higher level. The subtree for 'abcde' is the subtree of 'bcde' less any 'a' nodes. The tree of all solutions is notionally infinite, but the program prints one branch to a finite depth so only a finite part of the tree is evaluated thanks to lazy evaluation.

let rec
 even = lambda n. (n/2)*2=n,

repeated = lambda len. lambda s.
   { test:  ? s = s2;s2 ? }
   let rec
     r = lambda half. lambda s2.
       if half=len/2 then
         let rec
           r2 = lambda s1. lambda s2.
             if null s2 then true
             else if hd s1=hd s2 then
               r2  tl s1  tl s2
             else false
         in r2 s  tl s2
       else r (half+1) tl s2
   in r 1 s,

paths = lambda depth. lambda T.
   {paths from root of T to a given depth}
   let rec
     across = lambda d. lambda Ancestors.
              lambda T. lambda Rest.
       if null T then Rest
       else down d (hd T :: Ancestors) (hd tl T)
                   (across d Ancestors tl tl T Rest),
     down = lambda d. lambda Ancestors.
            lambda T. lambda Rest.
       if d >= depth then Ancestors :: Rest
       else {d < depth} across (d+1) Ancestors T Rest
   in across 1 nil T nil

in let                        { build does the work }
 build = lambda N.
   let rec
     T = toplevel 1,

toplevel = lambda M.
       if M > N then nil
       else M :: ((f 2 (M::nil) T) ::
                  (toplevel (M+1))),

f = lambda len. lambda seq. lambda Shadow.
       if null Shadow then nil
       else
       let
         others = f  len seq  tl tl Shadow,
         seq2 = hd Shadow :: seq
       in if even len and repeated len seq2 then others
          else (hd Shadow) ::
               ((f (len+1) seq2  hd tl Shadow)::others)
   in T

in hd( paths 20 ( build 3 ))  {print one}

There are more λ-calculus examples here.

References

[All89] Lloyd Allison, 'Circular Programs and Self-Referential Structures', Software Practice & Experience, 19(2), pp.99-109, doi:10.1002/spe.4380190202, arxiv:2403.01866, February 1989.
[All93] Lloyd Allison, 'Applications of Recursively Defined Data Structures', Australian Computer Journal, 25(1), pp.14-20, arxiv:2206.12795, 1993.
And other publications.