phi and the Rogers-Ramanujan identities

[Geoffrey Caveney]

The Rogers-Ramanujan identities are closely related to the Rogers-Ramanujan continued fraction, so perhaps it should not be surprising that there may be a connection between them and phi (the golden ratio).

However, in the standard sources, I do not see any comment or observation on the fact that the ratio of the number of partitions of n into parts 5k+1 or 5k+4 (A003114) to the number of partitions of n into parts 5k+2 or 5k+3 (A003106), as n approaches infinity, appears to approach the limit phi, albeit very slowly.

Also, this ratio appears to approach phi only as slowly increasing values less than phi, rather than as values alternately less than and greater than phi, as in the famous case of the ratios of successive terms of the Fibonacci sequence.

For example, even the ratio of the value of a(10000) of A003114 to the value of a(10000) of A003106, which is approximately 1.6154353662…, is only slightly closer to phi than 21/13, and not nearly as close to phi as 55/34.

Nevertheless, it seems evident from an examination of the ratios of the first 10000 terms of these sequences, that this ratio is approaching the limit phi, however slowly. 

This connection occurred to me due to an initial examination of Gérard’s sequences A035401-A035428. It seems possible to me that as X approaches infinity, the ratios of the numbers of terms with values less than X of certain pairs of these sequences may also be approaching phi, albeit perhaps even more slowly than the ratio described above. Consider for example pairs of sequences such as A035422/A035421 ; A035421/A035401 ; A035401/A035406 ; A035421/A035404 ; A035404/A035409 ; etc. I emphasize that this is an initial observation based on the very limited data available in the short sequences in their current state as Gérard gave them to OEIS.

Geoffrey