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NUMBER OF RATIONAL KNOTS AND LINKS


2n-4+2[n/2]-2




This very simple formula is probably the first combinatorial result in knot theory, giving the exact number of links belonging to a particular class, and not only its approximation.

The infinite sequences b(n) and d**(n) are discovered before, in some other context, and included in the book The Encyclopedia of Integer Sequences and On-Line Encyclopedia of Integer Sequences by N.J.A. Sloane. They are denoted, respectively, as A001224 and A005418, and related to "Binary grids" and "Packing a box with n dominoes".