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A336398 Number of rational knots (or two-bridge knots) with n crossings (chiral pairs counted as distinct). 4
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 (list; graph; refs; listen; history; text; internal format)
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 A137424

Adjacent sequences:  A336395 A336396 A336397 * A336399 A336400 A336401

KEYWORD

nonn,easy

AUTHOR

Andrey Zabolotskiy, Jul 20 2020

STATUS

approved

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Last modified October 7 16:02 EDT 2022. Contains 357275 sequences. (Running on oeis4.)