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

2,2

Links

Table of n, a(n) for n=2..33.

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).

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).

Prog

(Python) [(2**(n-2) + [-1, 2**(n//2), -1, 2**(n//2)+2][n%4])//3 for n in range(2, 30)]

Crossrefs

Cf. A018240, A090597, A329908.

Sequence in context: A155944 A350087 A091232 * A209337 A243004 A375156

Adjacent sequences: A336395 A336396 A336397 * A336399 A336400 A336401

Keyword

nonn,easy

Author

Andrey Zabolotskiy, Jul 20 2020

Status

approved