A336398 Number of rational knots (or two-bridge knots) with n crossings (chiral pairs counted as distinct).
0, 2, 1, 4, 5, 14, 21, 48, 85, 182, 341, 704, 1365, 2774, 5461, 11008, 21845, 43862, 87381, 175104, 349525, 699734, 1398101, 2797568, 5592405, 11187542, 22369621, 44744704, 89478485, 178967894, 357913941, 715849728
Offset: 2
Links
- C. Ernst and D. W. Sumners, The Growth of the Number of Prime Knots, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987 (see Theorem 1, formulas for TK_n^*).
- Taizo Kanenobu and Toshio Sumi, Polynomial Invariants of 2-Bridge Knots through 22 Crossings, Math. Comp. 60 (1993), 771-778, S17 (see Table 2).
- Index entries for linear recurrences with constant coefficients, signature (1,3,-1,0,-2,-4).
Programs
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Python
[(2**(n-2) + [-1, 2**(n//2), -1, 2**(n//2)+2][n%4])//3 for n in range(2, 30)]
Formula
(2^(n-2) - 1) / 3 if n is even,
(2^(n-2) + 2^((n-1)/2)) / 3 if n = 1 (mod 4),
(2^(n-2) + 2^((n-1)/2) + 2) / 3 if n = 3 (mod 4).
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 2*a(n-5) - 4*a(n-6).