Self-avoiding spiral chains

For chains with N links and in the limit N - oo, Blote and Hilhorst found the following results. 

For chains spiralling towards the outside, the total number of chains starting from an origin O is given by 

In the same way, for all double-spiral chains, the total number of chains is given 

These results are remarkable in particular they show that the number of selfavoiding spirals with N links increases only as an exponential of N1/2 whereas the number of self-avoiding chains increases as an exponential of N. Thus, we see from this example that the universality principles cannot be applied without discrimination and that one must be careful  

The number k is the constant which appears in the famous Hardy-Ramanujan formula (1917) which gives the number of partitions of an integer N (see Appendix B). This fact is not fortuitous and our readers shall see why in what follows. 

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