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For the linear equation

x[1] + x[2] + x[3] + x[4] + x[5] == n

Is there any efficient way to find all the non-negative integer solutions of this equation when n is large?

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FrobeniusSolve[ConstantArray[1, #], #2] &[5,4]

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For large n it would be better to use IntegerPartitions to avoid solutions that are identical up to permutation. Using the second and third arguments, IntegerPartitions[#, {5}, Range[0, #]] gives all the solutions {x[1],x[2],x[3],x[4],x[5]} with the structure x[1] >= x[2] >= x[3] >= x[4] >= x[5]

IntegerPartitions[#, {5}, Range[0, #]] &[12]

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  • $\begingroup$ @Wenzhe, my pleasure. Welcome to mma.se. $\endgroup$ – kglr May 10 '16 at 2:45
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Try this:

PadLeft[IntegerPartitions[n]]
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  • $\begingroup$ PadLeft[IntegerPartitions[12, 5]] $\endgroup$ – Bob Hanlon May 10 '16 at 3:14

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