Any help with this would be greatly appreciated!
I am trying to figure out how to PRECISELY determine the predecessor in a Fibonacci-LIKE sequence, not a TRUE Fibonacci sequence. I have attempted to use (1+sqrt(5n^2)-n)/2 but it only works for TRUE Fibonacci numbers. By Fibonacci-LIKE sequence, I mean ANY number that is NOT actually a true Fibonacci number but is in a similar additive sequence. For example, if I have the randomly chosen number "50896034545" and I want to determine what lesser number would precede it in a Fibonacci-LIKE sequence so that I could continually subtract each successive answer from the previous to form a sequence, I could simply divide by phi (1.618...) and then subtract the value from the given number and repeat.
HOWEVER, when I try to predict the successor value from the smallest value in the sequence so that I can repeatedly add the values to get back to my original number, it's never quite accurate enough in NON-TRUE Fibonacci sequences.
IN FACT, the inaccuracy is completely predictable and follows a saw tooth pattern of successively higher and lower value than it should predict which in itself is inaccurate by increasing values of Fibonacci-LIKE numbers.
What I'm asking is if there are more accurate predecessor and successor formulas for Fibonacci-LIKE sequences. That is not to say, more accurate values of phi. I've already tried phi to 1,500 digits and it's still the same problem.