Antireduce

Antireduce is an useful function in sequence pattern recognition. Antireduce takes a binary operation (usually the inverse of some other binary operation) and returns the result of applying it across a sequence of values:

(defn antireduce
  [op coll]

  (map
   (fn [i]
     (if (= i 0)
       (first coll)
       (op (nth coll i) (nth coll (dec i)))))
   (range (count coll))))

For example, we can apply this function to the triangular-numbers:

(= (take 4 (iterate (partial antireduce -) 
  [0 1 3 6 10 15 21 28 36 45])
  
  (0 1 3 6 10 15 21 28 36 45)
  (0 1 2 3 4 5 6 7 8 9)
  (0 1 1 1 1 1 1 1 1 1)
  (0 1 0 0 0 0 0 0 0 0))

Alternatively, we can use this on the factorial function by using the higher inverse-hyper-operator, division.

(= (take 2 (iterate (partial antireduce /) 
  [1 1 2 6 24 120 720 5040 40320 362880]))

  (1 1 2 6 24 120 720 5040 40320 362880)
  (1 1 2 3 4 5 6 7 8 9))

The antireduce function itself can be reversed:

(defn reducify
   [op coll]

   (map
    (fn [i]
      (reduce op (map (partial nth coll) (range (inc i)))))
    (range (count coll))))

This function should allow us to automatically recognise some basically sequences.