Starting from (1+x)^n = $C_0$ + $C_1$x + $C_2$$x^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.