Recently I formulated formula for pi that contains specific Lucas numbers of the first and the second kind and exponents of Mersenne prime numbers . Please see attachment.

For the first 50 odd exponents of Mersenne primes I get value: 3.141592655591489672… while the real value of pi rounded to 18 decimal digits is 3.141592653589793238

I would like to hear your opinion is it possible that formula is correct and what would be approach to prove it in that case or it is just interesting numerical coincidence.