A geometric sequence with a common ratio (or ratio constant) of 2 has the recurrence Sn=Sn-1 + Sn-1, where n stands for the index in the sequence S.
The Fibonacci sequence with a ratio constant of roughly 1.618 (known as Phi) has the recurrence Sn=Sn-1 + Sn-2.
The Narayana's Cows sequence (OEIS A000930) with a ratio constant of roughly 1.4656 (call it Moo or the bovine ratio) has the recurrence Sn=Sn-1 + Sn-3.
And so on.
(2^0)+1 = 2^(0+1)
(Phi^1)+1 = Phi^(1+1)
(Moo^2)+1 = Moo^(2+1)
Each of these equations simply shifts the bracket pairs along, so what's unique about the one featuring the golden ratio is that all the numerical values are 1's. Maybe that's what makes it beautiful.