Addition chains: Difference between revisions

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* brauer-chains(19) : count = 31 Ex: ( 1 2 3 4 8 11 19)
* brauer-chains(19) : count = 31 Ex: ( 1 2 3 4 8 11 19)
* non-brauer-chains(19) : count = 2 Ex: ( 1 2 3 6 7 12 19)
* non-brauer-chains(19) : count = 2 Ex: ( 1 2 3 6 7 12 19)

=={{header|C#|C sharp}}==
{{trans|Java}}
<lang csharp>using System;

namespace AdditionChains {
class Program {
static int[] Prepend(int n, int[] seq) {
int[] result = new int[seq.Length + 1];
Array.Copy(seq, 0, result, 1, seq.Length);
result[0] = n;
return result;
}

static Tuple<int, int> CheckSeq(int pos, int[] seq, int n, int min_len) {
if (pos > min_len || seq[0] > n) return new Tuple<int, int>(min_len, 0);
if (seq[0] == n) return new Tuple<int, int>(pos, 1);
if (pos < min_len) return TryPerm(0, pos, seq, n, min_len);
return new Tuple<int, int>(min_len, 0);
}

static Tuple<int, int> TryPerm(int i, int pos, int[] seq, int n, int min_len) {
if (i > pos) return new Tuple<int, int>(min_len, 0);

Tuple<int, int> res1 = CheckSeq(pos + 1, Prepend(seq[0] + seq[i], seq), n, min_len);
Tuple<int, int> res2 = TryPerm(i + 1, pos, seq, n, res1.Item1);

if (res2.Item1 < res1.Item1) return res2;
if (res2.Item1 == res1.Item1) return new Tuple<int, int>(res2.Item1, res1.Item2 + res2.Item2);

throw new Exception("TryPerm exception");
}

static Tuple<int, int> InitTryPerm(int x) {
return TryPerm(0, 0, new int[] { 1 }, x, 12);
}

static void FindBrauer(int num) {
Tuple<int, int> res = InitTryPerm(num);
Console.WriteLine();
Console.WriteLine("N = {0}", num);
Console.WriteLine("Minimum length of chains: L(n)= {0}", res.Item1);
Console.WriteLine("Number of minimum length Brauer chains: {0}", res.Item2);
}

static void Main(string[] args) {
int[] nums = new int[] { 7, 14, 21, 29, 32, 42, 64, 47, 79, 191, 382, 379 };
Array.ForEach(nums, n => FindBrauer(n));
}
}
}</lang>
{{out}}
<pre>N = 7
Minimum length of chains: L(n)= 4
Number of minimum length Brauer chains: 5

N = 14
Minimum length of chains: L(n)= 5
Number of minimum length Brauer chains: 14

N = 21
Minimum length of chains: L(n)= 6
Number of minimum length Brauer chains: 26

N = 29
Minimum length of chains: L(n)= 7
Number of minimum length Brauer chains: 114

N = 32
Minimum length of chains: L(n)= 5
Number of minimum length Brauer chains: 1

N = 42
Minimum length of chains: L(n)= 7
Number of minimum length Brauer chains: 78

N = 64
Minimum length of chains: L(n)= 6
Number of minimum length Brauer chains: 1

N = 47
Minimum length of chains: L(n)= 8
Number of minimum length Brauer chains: 183

N = 79
Minimum length of chains: L(n)= 9
Number of minimum length Brauer chains: 492

N = 191
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 7172

N = 382
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 4

N = 379
Minimum length of chains: L(n)= 12
Number of minimum length Brauer chains: 6583</pre>


=={{header|D}}==
=={{header|D}}==

Revision as of 23:39, 15 August 2018

Addition chains is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

An addition chain of length r for n is a sequence 1 = a(0) < a(1) < a(2) ... < a(r) = n , such as a(k) = a(i) + a(j) ( i < k and j < k , i may be = j) . Each member is the sum of two earlier members, not necessarily distincts.

A Brauer chain for n is an addition chain where a(k) = a(k-1) + a(j) with j < k. Each member uses the previous member as a summand.

We are interested in chains of minimal length L(n).

Task

For each n in {7,14,21,29,32,42,64} display the following : L(n), the count of Brauer chains of length L(n), an example of such a Brauer chain, the count of non-brauer chains of length L(n), an example of such a chain. (NB: counts may be 0 ).

Extra-credit: Same task for n in {47, 79, 191, 382 , 379, 12509}

References

  • OEIS sequences A079301, A079302. [1]
  • Richard K. Guy - Unsolved problems in Number Theory - C6 - Addition chains.

Example

  • minimal chain length l(19) = 6
  • brauer-chains(19) : count = 31 Ex: ( 1 2 3 4 8 11 19)
  • non-brauer-chains(19) : count = 2 Ex: ( 1 2 3 6 7 12 19)

C#

Translation of: Java

<lang csharp>using System;

namespace AdditionChains { 

   class Program {
       static int[] Prepend(int n, int[] seq) {
           int[] result = new int[seq.Length + 1];
           Array.Copy(seq, 0, result, 1, seq.Length);
           result[0] = n;
           return result;
       }

       static Tuple<int, int> CheckSeq(int pos, int[] seq, int n, int min_len) {
           if (pos > min_len || seq[0] > n) return new Tuple<int, int>(min_len, 0);
           if (seq[0] == n) return new Tuple<int, int>(pos, 1);
           if (pos < min_len) return TryPerm(0, pos, seq, n, min_len);
           return new Tuple<int, int>(min_len, 0);
       }

       static Tuple<int, int> TryPerm(int i, int pos, int[] seq, int n, int min_len) {
           if (i > pos) return new Tuple<int, int>(min_len, 0);

           Tuple<int, int> res1 = CheckSeq(pos + 1, Prepend(seq[0] + seq[i], seq), n, min_len);
           Tuple<int, int> res2 = TryPerm(i + 1, pos, seq, n, res1.Item1);

           if (res2.Item1 < res1.Item1) return res2;
           if (res2.Item1 == res1.Item1) return new Tuple<int, int>(res2.Item1, res1.Item2 + res2.Item2);

           throw new Exception("TryPerm exception");
       }

       static Tuple<int, int> InitTryPerm(int x) {
           return TryPerm(0, 0, new int[] { 1 }, x, 12);
       }

       static void FindBrauer(int num) {
           Tuple<int, int> res = InitTryPerm(num);
           Console.WriteLine();
           Console.WriteLine("N = {0}", num);
           Console.WriteLine("Minimum length of chains: L(n)= {0}", res.Item1);
           Console.WriteLine("Number of minimum length Brauer chains: {0}", res.Item2);
       }

       static void Main(string[] args) {
           int[] nums = new int[] { 7, 14, 21, 29, 32, 42, 64, 47, 79, 191, 382, 379 };
           Array.ForEach(nums, n => FindBrauer(n));
       }
   }

}</lang>

Output:
N = 7
Minimum length of chains: L(n)= 4
Number of minimum length Brauer chains: 5

N = 14
Minimum length of chains: L(n)= 5
Number of minimum length Brauer chains: 14

N = 21
Minimum length of chains: L(n)= 6
Number of minimum length Brauer chains: 26

N = 29
Minimum length of chains: L(n)= 7
Number of minimum length Brauer chains: 114

N = 32
Minimum length of chains: L(n)= 5
Number of minimum length Brauer chains: 1

N = 42
Minimum length of chains: L(n)= 7
Number of minimum length Brauer chains: 78

N = 64
Minimum length of chains: L(n)= 6
Number of minimum length Brauer chains: 1

N = 47
Minimum length of chains: L(n)= 8
Number of minimum length Brauer chains: 183

N = 79
Minimum length of chains: L(n)= 9
Number of minimum length Brauer chains: 492

N = 191
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 7172

N = 382
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 4

N = 379
Minimum length of chains: L(n)= 12
Number of minimum length Brauer chains: 6583

D

Translation of: Scala

<lang D>import std.stdio; import std.typecons;

alias Pair = Tuple!(int, int);

auto check_seq(int pos, int[] seq, int n, int min_len) { 

   if (pos>min_len || seq[0]>n) return Pair(min_len, 0);
   else if (seq[0] == n)        return Pair(    pos, 1);
   else if (pos<min_len)        return try_perm(0, pos, seq, n, min_len);
   else                         return Pair(min_len, 0);

}

auto try_perm(int i, int pos, int[] seq, int n, int min_len) { 

   if (i>pos) return Pair(min_len, 0);

   auto res1 = check_seq(pos+1, [seq[0]+seq[i]]~seq, n, min_len);
   auto res2 = try_perm(i+1, pos, seq, n, res1[0]);

   if (res2[0] < res1[0])       return res2;
   else if (res2[0] == res1[0]) return Pair(res2[0], res1[1]+res2[1]);
   else                         throw new Exception("Try_perm exception");

}

auto init_try_perm = function(int x) => try_perm(0, 0, [1], x, 12);

void find_brauer(int num) { 

   auto res = init_try_perm(num);
   writeln;
   writeln("N = ", num);
   writeln("Minimum length of chains: L(n)= ", res[0]);
   writeln("Number of minimum length Brauer chains: ", res[1]);

}

void main() { 

   auto nums = [7, 14, 21, 29, 32, 42, 64, 47, 79, 191, 382, 379];
   foreach (i; nums) {
       find_brauer(i);
   }

}</lang>

Output:
N = 7
Minimum length of chains: L(n)= 4
Number of minimum length Brauer chains: 5

N = 14
Minimum length of chains: L(n)= 5
Number of minimum length Brauer chains: 14

N = 21
Minimum length of chains: L(n)= 6
Number of minimum length Brauer chains: 26

N = 29
Minimum length of chains: L(n)= 7
Number of minimum length Brauer chains: 114

N = 32
Minimum length of chains: L(n)= 5
Number of minimum length Brauer chains: 1

N = 42
Minimum length of chains: L(n)= 7
Number of minimum length Brauer chains: 78

N = 64
Minimum length of chains: L(n)= 6
Number of minimum length Brauer chains: 1

N = 47
Minimum length of chains: L(n)= 8
Number of minimum length Brauer chains: 183

N = 79
Minimum length of chains: L(n)= 9
Number of minimum length Brauer chains: 492

N = 191
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 7172

N = 382
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 4

N = 379
Minimum length of chains: L(n)= 12
Number of minimum length Brauer chains: 6583

EchoLisp

<lang scheme>

2^n

(define exp2 (build-vector 32 (lambda(i)(expt 2 i))))

counters and results

(define-values (*minlg* *counts* *chains* *calls*) '(0 null null 0))

(define (register-hit chain lg ) (define idx (if (brauer? chain lg) 0 1)) 

   (when (< lg *minlg*) 
       (set! *counts* (make-vector 2 0))
       (set! *chains* (make-vector 2 ""))
       (set! *minlg* lg))
   (vector+= *counts* idx 1)
   (vector-set! *chains* idx (vector->list chain)))
is chain a brauer chain ?

(define (brauer? chain lg) 

   (for [(i (in-range 1 lg))]
       #:break (not (vector-search* (- [chain i] [chain (1- i)]) chain)) => #f
       #t))
all min chains to target n (brute force)

(define (chains n chain lg (a) (top) (tops null)) (++ *calls*) (set! top [chain lg]) 

   (cond 
   [(> lg *minlg*) #f] ;; too long
   [(= n top) (register-hit chain lg)]  ;; hit 
   [(< n top) #f] ;; too big
   [(and (< *minlg* 32) (< (* top [exp2 (- *minlg* lg)]) n)) #f] ;; too small
   [else
   (for*  ([i (in-range lg -1 -1)] [j (in-range lg (1- i) -1)])      
         (set! a (+ [chain i] [chain j]))
         #:continue (<= a top) ;; increasing sequence
         #:continue (memq a tops) ;; prevent duplicates
         (set! tops (cons a tops))
         (vector-push chain a)
         (chains n chain  (1+ lg))
         (vector-pop chain))]))

(define (task n) 

   (set!-values (*minlg* *calls*) '(Infinity 0 ))
   (chains n (make-vector 1 1) 0)
   (printf "L(%d) = %d - brauer-chains: %d  non-brauer: %d  chains: %a %a " 
        n *minlg* [*counts* 0] [*counts* 1] [*chains* 0] [*chains* 1]))

</lang>

Output:
(for-each task {7 14 21 29 32 42 64})

L(7) = 4 - brauer-chains: 5 non-brauer: 0 chains: (1 2 3 4 7)  
L(14) = 5 - brauer-chains: 14 non-brauer: 0 chains: (1 2 3 4 7 14)  
L(21) = 6 - brauer-chains: 26 non-brauer: 3 chains: (1 2 3 4 7 14 21) (1 2 4 5 8 13 21) 
L(29) = 7 - brauer-chains: 114 non-brauer: 18 chains: (1 2 3 4 7 11 18 29) (1 2 3 6 9 11 18 29) 
L(32) = 5 - brauer-chains: 1 non-brauer: 0 chains: (1 2 4 8 16 32)  
L(42) = 7 - brauer-chains: 78 non-brauer: 6 chains: (1 2 3 4 7 14 21 42) (1 2 4 5 8 13 21 42) 
L(64) = 6 - brauer-chains: 1 non-brauer: 0 chains: (1 2 4 8 16 32 64) 

;; a few extras
(task 47)
L(47) = 8 - brauer-chains: 183 non-brauer: 37 chains: (1 2 3 4 7 10 20 27 47) (1 2 3 5 7 14 19 28 47) 
(task 79)
L(79) = 9 - brauer-chains: 492 non-brauer: 129 chains: (1 2 3 4 7 9 18 36 43 79) (1 2 3 5 7 12 24 31 48 79)

Java

Translation of: D

<lang Java>public class AdditionChains { 

   private static class Pair {
       int f, s;

       Pair(int f, int s) {
           this.f = f;
           this.s = s;
       }
   }

   private static int[] prepend(int n, int[] seq) {
       int[] result = new int[seq.length + 1];
       result[0] = n;
       System.arraycopy(seq, 0, result, 1, seq.length);
       return result;
   }

   private static Pair check_seq(int pos, int[] seq, int n, int min_len) {
       if (pos > min_len || seq[0] > n) return new Pair(min_len, 0);
       else if (seq[0] == n) return new Pair(pos, 1);
       else if (pos < min_len) return try_perm(0, pos, seq, n, min_len);
       else return new Pair(min_len, 0);
   }

   private static Pair try_perm(int i, int pos, int[] seq, int n, int min_len) {
       if (i > pos) return new Pair(min_len, 0);

       Pair res1 = check_seq(pos + 1, prepend(seq[0] + seq[i], seq), n, min_len);
       Pair res2 = try_perm(i + 1, pos, seq, n, res1.f);

       if (res2.f < res1.f) return res2;
       else if (res2.f == res1.f) return new Pair(res2.f, res1.s + res2.s);
       else throw new RuntimeException("Try_perm exception");
   }

   private static Pair init_try_perm(int x) {
       return try_perm(0, 0, new int[]{1}, x, 12);
   }

   private static void find_brauer(int num) {
       Pair res = init_try_perm(num);
       System.out.println();
       System.out.println("N = " + num);
       System.out.println("Minimum length of chains: L(n)= " + res.f);
       System.out.println("Number of minimum length Brauer chains: " + res.s);
   }

   public static void main(String[] args) {
       int[] nums = new int[]{7, 14, 21, 29, 32, 42, 64, 47, 79, 191, 382, 379};
       for (int i : nums) {
           find_brauer(i);
       }
   }

}</lang>

Output:
N = 7
Minimum length of chains: L(n)= 4
Number of minimum length Brauer chains: 5

N = 14
Minimum length of chains: L(n)= 5
Number of minimum length Brauer chains: 14

N = 21
Minimum length of chains: L(n)= 6
Number of minimum length Brauer chains: 26

N = 29
Minimum length of chains: L(n)= 7
Number of minimum length Brauer chains: 114

N = 32
Minimum length of chains: L(n)= 5
Number of minimum length Brauer chains: 1

N = 42
Minimum length of chains: L(n)= 7
Number of minimum length Brauer chains: 78

N = 64
Minimum length of chains: L(n)= 6
Number of minimum length Brauer chains: 1

N = 47
Minimum length of chains: L(n)= 8
Number of minimum length Brauer chains: 183

N = 79
Minimum length of chains: L(n)= 9
Number of minimum length Brauer chains: 492

N = 191
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 7172

N = 382
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 4

N = 379
Minimum length of chains: L(n)= 12
Number of minimum length Brauer chains: 6583

Kotlin

As far as the minimal Brauer chains are concerned, I've translated the code in the Scala entry which even on my modest machine is reasonably fast for generating these in isolation - negligible for N <= 79, 10 seconds for N = 191, 25 seconds for N = 382 and about 2.5 minutes for N = 379. However, N = 12509 (which according to tables requires a minimum length of 17) is still well out of reach using this code.

I've then extended the code to count the number of non-Brauer chains of the same minimum length - basically 'brute' force to generate all addition chains and then subtracted the number of Brauer ones - plus examples for both. For N <= 64 this adds little to the execution time but adds about 1 minute for N = 79 and I gave up waiting for N = 191! To deal with these glacial execution times, I've added code which enables you to suppress the non-Brauer generation for N above a specified figure. <lang scala>// version 1.1.51

var example: List<Int>? = null

fun checkSeq(pos: Int, seq: List<Int>, n: Int, minLen: Int): Pair<Int, Int> = 

   if (pos > minLen || seq[0] > n) minLen to 0
   else if (seq[0] == n)           { example = seq; pos to 1 }
   else if (pos < minLen)          tryPerm(0, pos, seq, n, minLen)
   else                            minLen to 0

fun tryPerm(i: Int, pos: Int, seq: List<Int>, n: Int, minLen: Int): Pair<Int, Int> { 

   if (i > pos) return minLen to 0
   val res1 = checkSeq(pos + 1, listOf(seq[0] + seq[i]) + seq, n, minLen)
   val res2 = tryPerm(i + 1, pos, seq, n, res1.first)
   return if (res2.first < res1.first)       res2
          else if (res2.first == res1.first) res2.first to (res1.second + res2.second)
          else                               { println("Exception in tryPerm"); 0 to 0 }

}

fun initTryPerm(x: Int, minLen: Int) = tryPerm(0, 0, listOf(1), x, minLen)

fun findBrauer(num: Int, minLen: Int, nbLimit: Int) { 

   val (actualMin, brauer) = initTryPerm(num, minLen)
   println("\nN = $num")
   println("Minimum length of chains : L($num) = $actualMin")
   println("Number of minimum length Brauer chains : $brauer")
   if (brauer > 0) println("Brauer example : ${example!!.reversed()}")
   example = null
   if (num <= nbLimit) {
       val nonBrauer = findNonBrauer(num, actualMin + 1, brauer)
       println("Number of minimum length non-Brauer chains : $nonBrauer")
       if (nonBrauer > 0) println("Non-Brauer example : ${example!!}")
       example = null
   }
   else {
       println("Non-Brauer analysis suppressed")
   }

}

fun isAdditionChain(a: IntArray): Boolean { 

   for (i in 2 until a.size) {
       if (a[i] > a[i - 1] * 2) return false
       var ok = false
       jloop@ for (j in i - 1 downTo 0) {
           for (k in j downTo 0) {
              if (a[j] + a[k] == a[i]) { ok = true; break@jloop }
           }
       }
       if (!ok) return false
   }
   if (example == null && !isBrauer(a)) example = a.toList()
   return true

}

fun isBrauer(a: IntArray): Boolean { 

   for (i in 2 until a.size) {
       var ok = false
       for (j in i - 1 downTo 0) {
           if (a[i - 1] + a[j] == a[i]) { ok = true; break }
       }
       if (!ok) return false
   }
   return true

}

fun findNonBrauer(num: Int, len: Int, brauer: Int): Int { 

   val seq = IntArray(len)
   seq[0] = 1
   seq[len - 1] = num
   for (i in 1 until len - 1) seq[i] = seq[i - 1] + 1
   var count = if (isAdditionChain(seq)) 1 else 0

   fun nextChains(index: Int) {
       while (true) {
           if (index < len - 1) nextChains(index + 1)
           if (seq[index] + len - 1 - index >= seq[len - 1]) return
           seq[index]++
           for (i in index + 1 until len - 1) seq[i] = seq[i - 1] + 1
           if (isAdditionChain(seq)) count++
       }
   }

   nextChains(2)
   return count - brauer

}

fun main(args: Array<String>) { 

   val nums = listOf(7, 14, 21, 29, 32, 42, 64, 47, 79, 191, 382, 379)
   println("Searching for Brauer chains up to a minimum length of 12:")
   for (num in nums) findBrauer(num, 12, 79)

}</lang>

Output:
Searching for Brauer chains up to a minimum length of 12:

N = 7
Minimum length of chains : L(7) = 4
Number of minimum length Brauer chains : 5
Brauer example : [1, 2, 3, 4, 7]
Number of minimum length non-Brauer chains : 0

N = 14
Minimum length of chains : L(14) = 5
Number of minimum length Brauer chains : 14
Brauer example : [1, 2, 3, 4, 7, 14]
Number of minimum length non-Brauer chains : 0

N = 21
Minimum length of chains : L(21) = 6
Number of minimum length Brauer chains : 26
Brauer example : [1, 2, 3, 4, 7, 14, 21]
Number of minimum length non-Brauer chains : 3
Non-Brauer example : [1, 2, 4, 5, 8, 13, 21]

N = 29
Minimum length of chains : L(29) = 7
Number of minimum length Brauer chains : 114
Brauer example : [1, 2, 3, 4, 7, 11, 18, 29]
Number of minimum length non-Brauer chains : 18
Non-Brauer example : [1, 2, 3, 6, 9, 11, 18, 29]

N = 32
Minimum length of chains : L(32) = 5
Number of minimum length Brauer chains : 1
Brauer example : [1, 2, 4, 8, 16, 32]
Number of minimum length non-Brauer chains : 0

N = 42
Minimum length of chains : L(42) = 7
Number of minimum length Brauer chains : 78
Brauer example : [1, 2, 3, 4, 7, 14, 21, 42]
Number of minimum length non-Brauer chains : 6
Non-Brauer example : [1, 2, 4, 5, 8, 13, 21, 42]

N = 64
Minimum length of chains : L(64) = 6
Number of minimum length Brauer chains : 1
Brauer example : [1, 2, 4, 8, 16, 32, 64]
Number of minimum length non-Brauer chains : 0

N = 47
Minimum length of chains : L(47) = 8
Number of minimum length Brauer chains : 183
Brauer example : [1, 2, 3, 4, 7, 10, 20, 27, 47]
Number of minimum length non-Brauer chains : 37
Non-Brauer example : [1, 2, 3, 5, 7, 14, 19, 28, 47]

N = 79
Minimum length of chains : L(79) = 9
Number of minimum length Brauer chains : 492
Brauer example : [1, 2, 3, 4, 7, 9, 18, 36, 43, 79]
Number of minimum length non-Brauer chains : 129
Non-Brauer example : [1, 2, 3, 5, 7, 12, 24, 31, 48, 79]

N = 191
Minimum length of chains : L(191) = 11
Number of minimum length Brauer chains : 7172
Brauer example : [1, 2, 3, 4, 7, 8, 15, 22, 44, 88, 103, 191]
Non-Brauer analysis suppressed

N = 382
Minimum length of chains : L(382) = 11
Number of minimum length Brauer chains : 4
Brauer example : [1, 2, 4, 5, 9, 14, 23, 46, 92, 184, 198, 382]
Non-Brauer analysis suppressed

N = 379
Minimum length of chains : L(379) = 12
Number of minimum length Brauer chains : 6583
Brauer example : [1, 2, 3, 4, 7, 10, 17, 27, 44, 88, 176, 203, 379]
Non-Brauer analysis suppressed

Scala

Following Scala implementation finds number of minimum length Brauer chains and corresponding length. <lang Scala> object chains{ 

   def check_seq(pos:Int,seq:List[Int],n:Int,min_len:Int):(Int,Int) = {
       if(pos>min_len || seq(0)>n)             (min_len,0)
       else if(seq(0) == n)                    (pos,1)
       else if(pos<min_len)                    try_perm(0,pos,seq,n,min_len)
       else                                    (min_len,0)
   }
   
   def try_perm(i:Int,pos:Int,seq:List[Int],n:Int,min_len:Int):(Int,Int) = {
       if(i>pos)           return (min_len,0)
       val res1 = check_seq(pos+1,seq(0)+seq(i) :: seq,n,min_len)
       val res2 = try_perm(i+1,pos,seq,n,res1._1)
       if(res2._1 < res1._1)                   res2
       else if(res2._1 == res1._1)             (res2._1,res1._2 + res2._2)
       else {
           println("Try_perm exception")
           (0,0)
       }
   }
   val init_try_perm = (x:Int) => try_perm(0,0,List[Int](1),x,10)
   def find_brauer(num:Int): Unit = {
       val res = init_try_perm(num)
       println()
       println("N = %d".format(num))
       println("Minimum length of chains: L(n)= " + res._1 + f"\nNumber of minimum length Brauer chains: " + res._2)
   }
   def main(args:Array[String]) :Unit = {
       val nums = List(7,14,21,29,32,42,64)
       for (i <- nums)     find_brauer(i)
   }

} </lang> 

N = 7
Minimum length of chains: L(n)= 4
Number of minimum length Brauer chains: 5

N = 14
Minimum length of chains: L(n)= 5
Number of minimum length Brauer chains: 14

N = 21
Minimum length of chains: L(n)= 6
Number of minimum length Brauer chains: 26

N = 29
Minimum length of chains: L(n)= 7
Number of minimum length Brauer chains: 114

N = 32
Minimum length of chains: L(n)= 5
Number of minimum length Brauer chains: 1

N = 42
Minimum length of chains: L(n)= 7
Number of minimum length Brauer chains: 78

N = 64
Minimum length of chains: L(n)= 6
Number of minimum length Brauer chains: 1
N = 47
Minimum length of chains: L(n)= 8
Number of minimum length Brauer chains: 183

N = 79
Minimum length of chains: L(n)= 9
Number of minimum length Brauer chains: 492

N = 191
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 7172

N = 191
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 7172

N = 382
Minimum length of chains: L(n)= 11
Number of minimum length Brauer chains: 4

N = 379
Minimum length of chains: L(n)= 12
Number of minimum length Brauer chains: 6583

zkl

Translation of: EchoLisp

<lang zkl>var exp2=(32).pump(List,(2).pow), // 2^n, n=0..31 

   _minlg, _counts, _chains;      // counters and results

fcn register_hit(chain,lg){ // save [upto 2] chains 

  idx:=(if(isBrauer(chain,lg)) 0 else 1);
  if(lg<_minlg) _counts,_chains,_minlg=List(0,0), List("",""), lg;
  _counts[idx]+=1;
  _chains[idx]=chain.copy();

} 

   // is chain a brauer chain ?

fcn isBrauer(chain,lg){ 

  foreach i in (lg){
     if(not chain.holds(chain[i+1] - chain[i])) return(False);
   }
   True

} 

   // all min chains to target n (brute force)

fcn chains(n,chain,lg){ 

  top,tops:=chain[lg], List();
  if(lg>_minlg)   {}			   // too long
  else if(n==top) register_hit(chain,lg); // hit 
  else if(n<top)  {}			   // too big
  else if((_minlg<32) and (top*exp2[_minlg - lg]<n)){} // too small
  else{
     foreach i,j in ([lg..0,-1],[lg..i,-1]){
        a:=chain[i] + chain[j];

if(a<=top) continue; // increasing sequence if(tops.holds(a)) continue; // prevent duplicates tops.append(a); chain.append(a); self.fcn(n,chain,lg+1); // recurse chain.pop(); 

     }
  }

}</lang> <lang zkl>fcn task(n){ 

  _minlg=(0).MAX;
  chains(n,List(1),0);
  println("L(%2d) = %d; Brauer-chains: %3d; non-brauer: %3d; chains: %s"
        .fmt(n,_minlg,_counts.xplode(),_chains.filter()));

} T(7,14,21,29,32,42,64,47,79).apply2(task);</lang>

Output:
L( 7) = 4; Brauer-chains:   5; non-brauer:   0; chains: L(L(1,2,3,4,7))
L(14) = 5; Brauer-chains:  14; non-brauer:   0; chains: L(L(1,2,3,4,7,14))
L(21) = 6; Brauer-chains:  26; non-brauer:   3; chains: L(L(1,2,3,4,7,14,21),L(1,2,4,5,8,13,21))
L(29) = 7; Brauer-chains: 114; non-brauer:  18; chains: L(L(1,2,3,4,7,11,18,29),L(1,2,3,6,9,11,18,29))
L(32) = 5; Brauer-chains:   1; non-brauer:   0; chains: L(L(1,2,4,8,16,32))
L(42) = 7; Brauer-chains:  78; non-brauer:   6; chains: L(L(1,2,3,4,7,14,21,42),L(1,2,4,5,8,13,21,42))
L(64) = 6; Brauer-chains:   1; non-brauer:   0; chains: L(L(1,2,4,8,16,32,64))
L(47) = 8; Brauer-chains: 183; non-brauer:  37; chains: L(L(1,2,3,4,7,10,20,27,47),L(1,2,3,5,7,14,19,28,47))
L(79) = 9; Brauer-chains: 492; non-brauer: 129; chains: L(L(1,2,3,4,7,9,18,36,43,79),L(1,2,3,5,7,12,24,31,48,79))
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