Dreary Design (Large1)

A color is a triple of component values giving the levels of red, green, and blue. Each component is an integer from $0$ to $K$, inclusive. For example, in the color system with $K = 3$, the colors $(0, 2, 3)$ and $(0, 3, 2)$ are two of the distinct colors available.

A color is bland if every pair of its component values differs by at most $V$. For example, in a system with $K = 2$ and $V = 1$, the color $(2, 1, 1)$ is bland: the red and green components differ by $1$, the red and blue components differ by $1$, the green and blue components differ by $0$, and none of those differences exceeds $1$. The color $(2, 1, 0)$ is not bland, because its red and blue components differ by more than $1$.

Mr. Turner paints gloomy landscapes and wants to design a color system with many bland colors in it. Given $K$ and $V$, count the distinct bland colors.

The first line contains the number of test cases $T$. Each of the next $T$ lines contains two integers $K$ and $V$ separated by a space.

Limits

• $1 \le T \le 100$
• $0 \le K \le 2555$
• $0 \le V \le 555$
• $V \le K$
• Every answer fits in a signed 32-bit integer.

For each test case, print one line of the form Case #x: y, where x is the test case number starting from $1$ and y is the number of distinct bland colors.

With $K = 1$ and $V = 1$ there are eight possible colors: $(0, 0, 0)$, $(0, 0, 1)$, $(0, 1, 0)$, $(0, 1, 1)$, $(1, 0, 0)$, $(1, 0, 1)$, $(1, 1, 0)$, and $(1, 1, 1)$. All eight are bland, because no two components differ by more than $1$.

With $K = 1$ and $V = 0$ the same eight colors are possible, but only $(0, 0, 0)$ and $(1, 1, 1)$ are bland.