Starting from (1+x)^n = C_0 + C_1x + C_2x^2 + ... + C_n*x^n:

x = 1: C_0 + C_1 + C_2 + ... + C_n = 2^n x = -1: C_0 - C_1 + C_2 - ... + (-1)^n*C_n = 0 Adding: C_0 + C_2 + C_4 + ... = 2^{n-1} Subtracting: C_1 + C_3 + C_5 + ... = 2^{n-1}

x = i (imaginary unit): Separating real and imaginary parts gives sums of coefficients at indices differing by 4.

x = omega (cube root of unity): Combined with x = 1 and x = omega^2, gives sums of coefficients at indices differing by 3.