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).

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: Kotlin
`#include <stdio.h>#include <stdlib.h>#include <string.h> #define TRUE 1#define FALSE 0 typedef int bool; typedef struct {    int x, y;} pair; int* example = NULL;int exampleLen = 0; void reverse(int s[], int len) {    int i, j, t;    for (i = 0, j = len - 1; i < j; ++i, --j) {        t = s[i];        s[i] = s[j];        s[j] = t;    }} pair tryPerm(int i, int pos, int seq[], int n, int len, int minLen); pair checkSeq(int pos, int seq[], int n, int len, int minLen) {    pair p;    if (pos > minLen || seq[0] > n) {        p.x = minLen; p.y = 0;        return p;    }    else if (seq[0] == n) {        example = malloc(len * sizeof(int));        memcpy(example, seq, len * sizeof(int));        exampleLen = len;        p.x = pos; p.y = 1;        return p;    }    else if (pos < minLen) {        return tryPerm(0, pos, seq, n, len, minLen);    }    else {        p.x = minLen; p.y = 0;        return p;    }} pair tryPerm(int i, int pos, int seq[], int n, int len, int minLen) {    int *seq2;    pair p, res1, res2;    size_t size = sizeof(int);        if (i > pos) {        p.x = minLen; p.y = 0;        return p;    }    seq2 = malloc((len + 1) * size);    memcpy(seq2 + 1, seq, len * size);    seq2[0] = seq[0] + seq[i];    res1 = checkSeq(pos + 1, seq2, n, len + 1, minLen);    res2 = tryPerm(i + 1, pos, seq, n, len, res1.x);    free(seq2);    if (res2.x < res1.x)        return res2;    else if (res2.x == res1.x) {        p.x = res2.x; p.y = res1.y + res2.y;        return p;    }    else {        printf("Error in tryPerm\n");        p.x = 0; p.y = 0;        return p;    }} pair initTryPerm(int x, int minLen) {    int seq[1] = {1};    return tryPerm(0, 0, seq, x, 1, minLen);} void printArray(int a[], int len) {    int i;    printf("[");    for (i = 0; i < len; ++i) printf("%d ", a[i]);    printf("\b]\n");} bool isBrauer(int a[], int len) {    int i, j;    bool ok;    for (i = 2; i < len; ++i) {        ok = FALSE;        for (j = i - 1; j >= 0; j--) {            if (a[i-1] + a[j] == a[i]) {                ok = TRUE;                break;            }        }        if (!ok) return FALSE;    }    return TRUE;} bool isAdditionChain(int a[], int len) {    int i, j, k;    bool ok, exit;    for (i = 2; i < len; ++i) {        if (a[i] > a[i - 1] * 2) return FALSE;        ok = FALSE; exit = FALSE;        for (j = i - 1; j >= 0; --j) {            for (k = j; k >= 0; --k) {               if (a[j] + a[k] == a[i]) { ok = TRUE; exit = TRUE; break; }            }            if (exit) break;        }        if (!ok) return FALSE;    }    if (example == NULL && !isBrauer(a, len)) {        example = malloc(len * sizeof(int));        memcpy(example, a, len * sizeof(int));        exampleLen = len;    }    return TRUE;} void nextChains(int index, int len, int seq[], int *pcount) {    for (;;) {        int i;        if (index < len - 1) {           nextChains(index + 1, len, seq, pcount);        }        if (seq[index] + len - 1 - index >= seq[len - 1]) return;        seq[index]++;        for (i = index + 1; i < len - 1; ++i) {            seq[i] = seq[i-1] + 1;        }        if (isAdditionChain(seq, len)) (*pcount)++;    }} int findNonBrauer(int num, int len, int brauer) {    int i, count = 0;    int *seq = malloc(len * sizeof(int));    seq[0] = 1;    seq[len - 1] = num;    for (i = 1; i < len - 1; ++i) {        seq[i] = seq[i - 1] + 1;    }    if (isAdditionChain(seq, len)) count = 1;    nextChains(2, len, seq, &count);    free(seq);    return count - brauer;} void findBrauer(int num, int minLen, int nbLimit) {    pair p = initTryPerm(num, minLen);    int actualMin = p.x, brauer = p.y, nonBrauer;    printf("\nN = %d\n", num);    printf("Minimum length of chains : L(%d) = %d\n", num, actualMin);    printf("Number of minimum length Brauer chains : %d\n", brauer);    if (brauer > 0) {        printf("Brauer example : ");        reverse(example, exampleLen);        printArray(example, exampleLen);    }    if (example != NULL) {        free(example);        example = NULL;         exampleLen = 0;    }    if (num <= nbLimit) {        nonBrauer = findNonBrauer(num, actualMin + 1, brauer);        printf("Number of minimum length non-Brauer chains : %d\n", nonBrauer);        if (nonBrauer > 0) {            printf("Non-Brauer example : ");            printArray(example, exampleLen);        }        if (example != NULL) {            free(example);            example = NULL;             exampleLen = 0;        }    }    else {        printf("Non-Brauer analysis suppressed\n");    }} int main() {    int i;    int nums[12] = {7, 14, 21, 29, 32, 42, 64, 47, 79, 191, 382, 379};    printf("Searching for Brauer chains up to a minimum length of 12:\n");    for (i = 0; i < 12; ++i) findBrauer(nums[i], 12, 79);    return 0;}`
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
```

## C#

Translation of: Java
`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));        }    }}`
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
`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);    }}`
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

` ;; 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])) `
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
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)
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)
```

## Go

Translation of: Kotlin
`package main import "fmt" var example []int func reverse(s []int) {    for i, j := 0, len(s)-1; i < j; i, j = i+1, j-1 {        s[i], s[j] = s[j], s[i]    }} func checkSeq(pos, n, minLen int, seq []int) (int, int) {    switch {    case pos > minLen || seq[0] > n:        return minLen, 0    case seq[0] == n:        example = seq        return pos, 1    case pos < minLen:        return tryPerm(0, pos, n, minLen, seq)    default:        return minLen, 0    }} func tryPerm(i, pos, n, minLen int, seq []int) (int, int) {    if i > pos {        return minLen, 0    }    seq2 := make([]int, len(seq)+1)    copy(seq2[1:], seq)    seq2[0] = seq[0] + seq[i]    res11, res12 := checkSeq(pos+1, n, minLen, seq2)    res21, res22 := tryPerm(i+1, pos, n, res11, seq)    switch {    case res21 < res11:        return res21, res22    case res21 == res11:        return res21, res12 + res22    default:        fmt.Println("Error in tryPerm")        return 0, 0    }} func initTryPerm(x, minLen int) (int, int) {    return tryPerm(0, 0, x, minLen, []int{1})} func findBrauer(num, minLen, nbLimit int) {    actualMin, brauer := initTryPerm(num, minLen)    fmt.Println("\nN =", num)    fmt.Printf("Minimum length of chains : L(%d) = %d\n", num, actualMin)    fmt.Println("Number of minimum length Brauer chains :", brauer)    if brauer > 0 {        reverse(example)        fmt.Println("Brauer example :", example)    }    example = nil    if num <= nbLimit {        nonBrauer := findNonBrauer(num, actualMin+1, brauer)        fmt.Println("Number of minimum length non-Brauer chains :", nonBrauer)        if nonBrauer > 0 {            fmt.Println("Non-Brauer example :", example)        }        example = nil    } else {        println("Non-Brauer analysis suppressed")    }} func isAdditionChain(a []int) bool {    for i := 2; i < len(a); i++ {        if a[i] > a[i-1]*2 {            return false        }        ok := false    jloop:        for j := i - 1; j >= 0; j-- {            for k := j; k >= 0; k-- {                if a[j]+a[k] == a[i] {                    ok = true                    break jloop                }            }        }        if !ok {            return false        }    }    if example == nil && !isBrauer(a) {        example = make([]int, len(a))        copy(example, a)    }    return true} func isBrauer(a []int) bool {    for i := 2; i < len(a); i++ {        ok := false        for j := i - 1; j >= 0; j-- {            if a[i-1]+a[j] == a[i] {                ok = true                break            }        }        if !ok {            return false        }    }    return true} func nextChains(index, le int, seq []int, pcount *int) {    for {        if index < le-1 {            nextChains(index+1, le, seq, pcount)        }        if seq[index]+le-1-index >= seq[le-1] {            return        }        seq[index]++        for i := index + 1; i < le-1; i++ {            seq[i] = seq[i-1] + 1        }        if isAdditionChain(seq) {            (*pcount)++        }    }} func findNonBrauer(num, le, brauer int) int {    seq := make([]int, le)    seq[0] = 1    seq[le-1] = num    for i := 1; i < le-1; i++ {        seq[i] = seq[i-1] + 1    }    count := 0    if isAdditionChain(seq) {        count = 1    }    nextChains(2, le, seq, &count)    return count - brauer} func main() {    nums := []int{7, 14, 21, 29, 32, 42, 64, 47, 79, 191, 382, 379}    fmt.Println("Searching for Brauer chains up to a minimum length of 12:")    for _, num := range nums {        findBrauer(num, 12, 79)    }}`
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
```

## Java

Translation of: D
`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);        }    }}`
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.

`// 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)}`
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.

` 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)    }} `
```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
`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();      }   }}`
`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);`
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))
```