has 1-sqrt5 been derived out of fibonacci numbers themselves

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I know the algebraic formulas use the sqrt5+-1 to get phi so indirectly its related to the Fibonacci numbers, but can someone lead me to where it has been derived out of the numbers themselves in a consistent pattern without the algebra?

Here is what i did

n→∞lim fn+fn+1+fn+2/fn+3=1.23606

46368 + 75025 + 121393= 242,786

242,786/196418= 1.236067977476606
another

2178309+3524578+5702887=11,405,774

11,405,774/9227465=1.236067977499779

one more

9425114757512643212875125+96151855463018422468774568+155576970220531065681649693
=311,153,940,441,062,100,365,737,984

311,153,940,441,062,100,365,737,984/251728825683549488150424261=1.2360679774997895732704694785035509073486321