Hyperelliptic curves, continued fractions, and Somos sequences

In the early nineties Michael Somos asked for the inner meaning of the fact that the quadratic recursions

and

are respectively satisfied by the bi-directional sequences

surprisingly consisting entirely of integers (which incidentally, increase at a hefty clip: the log of their h-th term is O(h²) as h tends to infinity.

In the meantime I had found quite independent reason for studying the continued fraction expansion of square roots of polynomials. Nonetheless, I found that those expansions inter alia produce sequences of integers satisfying quadratic recursions of Somos type. For instance, the sequence (C_h) [called 4-Somos by some of its friends] reports the denominators of a sequence of points on the quartic curve Y²=(X²-3)²+4(X-2), equivalently of the points M+hS, with M=(-1,1), S=(0,0) on the cubic curve V²-V=U³+3U²+2U. Indeed all Somos 4 and Somos 5 sequences are `elliptic sequences' in such a sense.

I will tell the story from very first principles and will mention generalisations. Thus, again for instance, the sequence

arises from adding multiples of the class of the divisor at infinity on the Jacobian of the curve Y²=(X³-4X+1)²+4(X-2) of genus 2 to the class of the divisor given by [(φ,0),( φ ,0)]; here, no doubt to the joy of adherents to the cult of Fibonacci, φ denotes the golden ratio.