OFFSET
2,2
LINKS
Paolo Xausa, Table of n, a(n) for n = 2..1000
C. Ernst and D. W. Sumners, The Growth of the Number of Prime Knots, Math. Proc. Cambridge Philos. Soc., Vol. 102, Iss. 2 (1987), pp. 303-315 (see Theorem 1, formulas for TK_n^*).
Taizo Kanenobu and Toshio Sumi, Polynomial Invariants of 2-Bridge Knots through 22 Crossings, Math. Comp., Vol. 60, No. 202 (1993), pp. 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).
G.f.: x^3 * (2-x-3*x^2-2*x^4) / ((1-x-2*x^2) * (1-x^2-2*x^4)). - Elmo R. Oliveira, Apr 12 2026
MATHEMATICA
LinearRecurrence[{1, 3, -1, 0, -2, -4}, {0, 2, 1, 4, 5, 14}, 50] (* Paolo Xausa, Apr 16 2026 *)
PROG
(Python) [(2**(n-2) + [-1, 2**(n//2), -1, 2**(n//2)+2][n%4])//3 for n in range(2, 30)]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Andrei Zabolotskii, Jul 20 2020
EXTENSIONS
More terms from Elmo R. Oliveira, Apr 12 2026
STATUS
approved
