In this paper I discuss a pattern observable in the n and P_n values of the first nineteen pancake graphs. This pattern could potentially hint at the as yet unknown diameters of the next graphs, and is at least viable for first seventy-four values of n, since it predicts values, in the aforementioned range, of P_n which fall between published lower and upper bounds. The pattern arises in sets of adjacent n's with equal differences between n and P_n, and is equivalent to a subsequence of the Fibonacci sequence. If one takes the known values of P_n and deletes n from each, one gets a difference value, which we call d, which allows one to arrange the numbers into corresponding blocks, so that the first block has 2 columns, the second 3, then 5 and then 8, and there a Fibonacci subsqeuence appears to be emerging (...2,3,5,8...). In this paper, I provide a formula for P_n for those n that follow this pattern:h(n) = n + ⌈(log_φ (-(n - 1)(√5 - √5φ) + φ³)) - 4⌉ - 1 , and test it against published upper and lower bounds for P_n for n ≤ 10000.