No of integral solutions with restrictions

Question

x+y+z+w=n,where x<=w,y<=w. total number of non negative integral solutions of this equation?is there any closed form formula. what is the generalized formula across any number of variables?

Without more information this question cannot be answered. Nevertheless, if it is related to Mathematica programming I recommend to see this answer Finding the number of solutions to a diophantine equation — Artes, Dec 28 '15 at 20:31
An initial question to address (the lack of which is probably the reason this was closed): what are the specifics of the "generalized problem"? — Daniel Lichtblau, Dec 29 '15 at 16:51

Bill · Answer 1 · 2015-12-28T21:16:41.293

Treat n==0 separately, because Reduce doesn't parenthesize it in the same way, and let Reduce find the number of solutions.

Join[{1}, Table[Length[Reduce[x+y+z+w==n && 0<=x<=w && 0<=y<=w && 0<=z<=n &&
   0<=w<=n, {x, y, z, w}, Integers]], {n, 1, 12}]]

which returns

{1, 2, 5, 9, 15, 23, 34, 47, 64, 84, 108, 136, 169}

and OEIS almost instantly finds a formula, a generating function, etc.

Mathematica can also find the generating function for you, if you don't need the context and possible connections to prior work.

FindGeneratingFunction[{1, 2, 5, 9, 15, 23, 34, 47, 64, 84, 108, 136, 169}, x]

Attempting to generalize this with one more variable

Join[{1}, Table[Length[Reduce[v+x+y+z+w==n && 0<=v<=w && 0<=x<=w && 0<=y<=w &&
   0<=z<=n && 0<=w<=n, {v, x, y, z, w}, Integers]], {n, 1, 12}]]

returns

{1, 2, 6, 13, 24, 41, 67, 102, 150, 213, 294, 396, 523}

but this is not found in the OEIS. FindGeneratingFunction is also unsuccessful.

This perhaps hints that some bright individual might contemplate this, discover some underlying pattern, submit your work to the OEIS and leave a bit of knowledge for those who might follow along behind you. Or it is possible that I have made some mistake or not correctly generalized your question.

No of integral solutions with restrictions

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