Integers equal to each other or differing by 1 will be called "almost equal " within the contents of this problem.

In how many ways can 2011 be expressed as a sum of almost equal addends?

The order of the addends in the expression is immaterial.

(In reply to

Can you really.... by Charlie)

IMHO, grammatically **no** and mathematically **yes.**

Mathematicians usually extend meanings beyond the original definitions. like raising a number to a zero(or negative , or fractional) power, extracting a square root from a negative number , building a 7-dimensional "cube" etc.

Such extensions are only required not to inroduce contradictions in any previously defined operations.

Following this reasoning a sum of a single addend is the addend itself and a sum of zero addends is zero.

However, Charlie's remark should be addressed in the solution, so the proper answer should be either 2011, including the trivial case of one addend, or 2010, discounting this trivial case.

I trust this resolution satisfies everyone.