Third time's the charm

Finding a closed form for sums of binomial coefficients by residue class

This is a medium-difficulty puzzle that tests your ability to exploit symmetry and algebraic structure in binomial expansions. Rather than computing individual terms, you'll need to recognize a pattern that lets you extract every third coefficient in a single expression.

The key insight involves using roots of unity as a filtering mechanism. By evaluating the binomial expansion at carefully chosen complex numbers, you can construct a linear combination of expansions that cancels unwanted terms and leaves only the coefficients you need. This technique generalizes beyond binomial coefficients and is foundational in combinatorics and discrete mathematics.

• Roots of unity and their orthogonality properties
• Filtering coefficients by residue class modulo an integer
• Binomial theorem applied to complex arguments
• Recognizing when a sum collapses to a simple closed form