Algorithm to find whole number sequences summing to a specific value

Question

I need an efficient general algorithm to calculate all the sequences of N whole numbers that sum to a given whole number, M. There are (M+N-1)!/[M!*(N-1)!] such sequences. For example, if N = 3 and M = 4 then

(M+N-1)!/[M!*(N-1)!] = 15, and those 15 sequences are

(0,0,4), (0,1,3), (0,2,2), (0,3,1), (0,4,0),

(1,0,3), (1,1,2), (1,2,1), (1,3,0), (2,0,2),

(2,1,1), (2,2,0), (3,0,1), (3,1,0), (4,0,0).

Ideally the algorithm would be given the first of these sequences and would then generate each successive sequence from the previous one, but that is not a necessary requirement.

Answer 1

For your case you can use a recursive algorithm. By implementing a recursive approach to generate sequences.

Here's a Python implementation that takes N, M, and the current sequence as input and generates all possible sequences:

def generate_sequences(N, M, current_sequence=[]):

if N == 0:

if M == 0:

print(current_sequence)

return

for i in range(M + 1):

new_sequence = current_sequence + [i]

generate_sequences(N - 1, M - i, new_sequence)

# Example usage:

N = 3

M = 4

generate_sequences(N, M)

Hope this will answer your question. If you need more explanation let me know and I will give more detailed answer.