MathDB

(FRA 5)

Let

\alpha(n)

be the number of pairs

(x, y)

of integers such that

x+y = n, 0 \le y \le x

, and let

\beta(n)

be the number of triples

(x, y, z)

such that

x + y + z = n

and

0 \le z \le y \le x.

Find a simple relation between

\alpha(n)

and the integer part of the number

\frac{n+2}{2}

and the relation among

\beta(n), \beta(n -3)

and

\alpha(n).

Then evaluate

\beta(n)

as a function of the residue of

n

modulo

6

. What can be said about

\beta(n)

and

1+\frac{n(n+6)}{12}

? And what about

\frac{(n+3)^2}{6}

? Find the number of triples

(x, y, z)

with the property

x+ y+ z \le n, 0 \le z \le y \le x

as a function of the residue of

n

modulo

6.

What can be said about the relation between this number and the number

\frac{(n+6)(2n^2+9n+12)}{72}

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