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Hofstadter sequences
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(* | |
* Hofstadter sequences | |
* https://en.wikipedia.org/wiki/Hofstadter_sequence | |
*) | |
signature QUEUE = | |
sig | |
type 'a queue = 'a list * 'a list | |
val empty: 'a queue | |
val enqueue: 'a -> 'a queue -> 'a queue | |
val dequeue: 'a queue -> 'a * 'a queue | |
end | |
structure Queue :> QUEUE = | |
struct | |
type 'a queue = 'a list * 'a list | |
val empty = ([], []) | |
fun enqueue x (front, back) = (front, x :: back) | |
fun dequeue ([], []) = raise Empty | |
| dequeue (front, back) = | |
let | |
val (front', back') = | |
if null front | |
then (rev back, []) | |
else (front, back) | |
in | |
(hd front', (tl front', back')) | |
end | |
end | |
(* Figure-Figure sequences | |
* | |
* F(1) = 1 | |
* S(1) = 2 | |
* F(n) = F(n-1) + S(n-1) | |
* | |
* n 1 2 3 4 5 6 7 8 9 | |
* F 1 3 7 12 18 26 35 45 56 | |
* S 2 4 5 6 8 9 10 11 13 | |
* | |
*) | |
fun figure n = | |
let | |
fun figure' f s _ 1 = (f, s) | |
| figure' f s queue n = | |
let | |
val f' = f + s | |
val queue' = Queue.enqueue f' queue | |
val s' = s + 1 | |
val (m, queue'') = Queue.dequeue queue' | |
in | |
if s' = m | |
then figure' f' (s' + 1) queue'' (n - 1) | |
else figure' f' s' queue' (n - 1) | |
end | |
in | |
figure' 1 2 Queue.empty n | |
end | |
(* G sequence *) | |
fun g 0 = 0 | |
| g n = n - g (g (n - 1)) | |
(* H sequence *) | |
fun h 0 = 0 | |
| h n = n - h (h (h (n - 1))) | |
(* Female and Male sequences *) | |
fun f 0 = 1 | |
| f n = n - m (f (n - 1)) | |
and m 0 = 0 | |
| m n = n - f (m (n - 1)) | |
(* Q sequence *) | |
fun q 1 = 1 | |
| q 2 = 1 | |
| q n = q (n - q (n - 1)) + q (n - q (n - 2)) | |
(* U sequence *) | |
(* Hofstadter - Conway $10K sequence*) | |
fun a 1 = 1 | |
| a 2 = 1 | |
| a n = a (a (n - 1)) + a (n - a (n - 1)) |
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