A336398 -id:A336398 - cp's OEIS frontend

A018240 Number of rational knots (or two-bridge knots) with n crossings (up to mirroring).

1, 1, 2, 3, 7, 12, 24, 45, 91, 176, 352, 693, 1387, 2752, 5504, 10965, 21931, 43776, 87552, 174933, 349867, 699392, 1398784, 2796885, 5593771, 11186176, 22372352, 44741973, 89483947, 178962432, 357924864, 715838805, 1431677611, 2863333376, 5726666752

Offset: 3

Alexander Stoimenow (stoimeno(AT)math.toronto.edu)

			The a(7)=7 rational knots with 7 crossings are 7, 52, 43, 322, 313, 2212, 21112. All the rational knots are listed in A122495.

S. Jablan and R. Sazdanović, LinKnot: Knot Theory by Computer, World Scientific Press, 2007.

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
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 5, 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).
P.-V. Koseleff, D. Pecker, Conway polynomials of two-bridge links, arXiv:1011.5992 [math.GT], 2010-2012 (only version 1 contains tables).
P.-V. Koseleff, D. Pecker, On Alexander-Conway polynomials of two-bridge links, Journal of Symbolic Computation 68 (2015), 215-229.
A. Stoimenow, Generating functions, Fibonacci numbers and rational knots, Journal of Algebra, 310 (2007), 491-525.
Index entries for sequences related to knots
Index entries for linear recurrences with constant coefficients, signature (-1,5,5,-2,-2,-8,-8).

Cf. A122495, A005418, A002863, A086825, A089790.

Cf. A018240 = number of rational knots, A005418 = number of rational knots and links, A001045 = Jacobsthal sequence (the difference between the number of rational links and knots), A090597 = rational links with n crossings, A329908, A336398.

LinearRecurrence[{-1, 5, 5, -2, -2, -8, -8}, {1, 1, 2, 3, 7, 12, 24}, 50] (* Harvey P. Dale, Sep 03 2013 *)
CoefficientList[Series[(1 - 2 x^2 - x^3 - x^4)/((1 - 2 x) (1 + x) (1 - 2 x^2) (1 + x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2014 *)

Vec((1-2*x^2-x^3-x^4)*x^3/((1-2*x)*(1+x)*(1-2*x^2)*(1+x^2))+O(x^66)) \\ Joerg Arndt, Aug 07 2014

a(n) = - a(n-1) + 5*(a(n-2)+a(n-3)) - 2*(a(n-4)+a(n-5)) - 8*(a(n-6)+a(n-7)). [Originally contributed as a separate sequence entry by Thomas A. Gittings, Dec 11 2003; see Stoimenow, Corollary 5.1 for proof]

G.f.: (1-2*x^2-x^3-x^4)*x^3/((1-2*x)*(1+x)*(1-2*x^2)*(1+x^2)). - R. J. Mathar, Sep 08 2008

Edited by Andrey Zabolotskiy, Jun 18 2020

A090597 a(n) = - a(n-1) + 5(a(n-2) + a(n-3)) - 2(a(n-4) + a(n-5)) - 8(a(n-6) + a(n-7)).

Original entry on oeis.org

0, 1, 1, 3, 3, 8, 12, 27, 45, 96, 176, 363, 693, 1408, 2752, 5547, 10965, 22016, 43776, 87723, 174933, 350208, 699392, 1399467, 2796885, 5595136, 11186176, 22375083, 44741973, 89489408

Offset: 3

Thomas A. Gittings, Dec 11 2003

Arises from a conjecture about sequence of rational links with n crossings.
Conjecture derived from: s(n) = k(n) + l(n): definition of sum of rational knots (k) and links (l) s(n) = 6s(n-2) -8s(n-4): see A005418 (Jablan's observation) d(n) = d(n-2) + 2d(n-4): see A001045 (modified Jacobsthal sequence) l(n) = k(n-1) + d(n): conjecture.
a(n) is the number of rational (2-component) links. - Slavik Jablan, Dec 26 2003
Also yields the number of meanders, reduced by symmetry, on an n X 3 rectangle (see A200893). - Jon Wild, Nov 25 2011

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
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 5, formulas for TL_n).
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,0,-2,-4).

This is the difference between A005418 and A018240.
Cf. A018240 = sequence of rational knots, A005418 = number of rational knots and links, A001045 = Jacobsthal sequence, A329908, A336398.
Cf. A200893, and see the third column of the triangle read by rows there.

a090597 n = a090597_list !! (n-3)
a090597_list = [0,1,1,3,3,8,12] ++ zipWith (-)
   (drop 4 $ zipWith (-) (map (* 5) zs) (drop 2 a090597_list))
   (zipWith (+) (drop 2 $ map (* 2) zs) (map (* 8) zs))
   where zs = zipWith (+) a090597_list $ tail a090597_list
-- Reinhard Zumkeller, Nov 24 2011

f[x_] := (x-x^3-2x^4-3x^5) / (1-x-3x^2+x^3+2x^5+4x^6); CoefficientList[ Series[ f[x], {x, 0, 29}], x] (* Jean-François Alcover, Dec 06 2011 *)
J[n_] := (2^n - (-1)^n)/3; Table[(J[n - 3] + J[(n - If[OddQ[n], 3, 0])/2])/2 , {n, 3, 31}] (* David Scambler, Dec 13 2011 *)
LinearRecurrence[{1,3,-1,0,-2,-4},{0,1,1,3,3,8},30] (* Harvey P. Dale, Nov 12 2013 *)

a(n) = +a(n-1) +3*a(n-2) -a(n-3) -2*a(n-5) -4*a(n-6). - R. J. Mathar, Nov 23 2011
G.f.: -x^4*(-1+x^2+3*x^4+2*x^3) / ( (2*x-1)*(1+x)*(2*x^2-1)*(1+x^2) ). - R. J. Mathar, Nov 23 2011
a(n) = (J(n-3) + J((n-3)/2))/2 if n is odd; (J(n-3) + J(n/2))/2 if n is even, where J is the Jacobsthal number A001045. - David Scambler, Dec 12 2011

A329908 Number of oriented rational links with crossing number n.

Original entry on oeis.org

2, 2, 5, 6, 15, 24, 51, 90, 187, 352, 715, 1386, 2795, 5504, 11051, 21930, 43947, 87552, 175275, 349866, 700075, 1398784, 2798251, 5593770, 11188907, 22372352, 44747435, 89483946, 178973355, 357924864, 715860651, 1431677610, 2863377067, 5726666752, 11453377195

Offset: 2

Michel Marcus, Jul 07 2020

Yuanan Diao, Michael Finney, and Dawn Ray, The number of oriented rational links with a given deficiency number, arXiv:2007.02819 [math.GT], 2020. See Theorem 3 p.9 and Table 1 p. 14.
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 TL_n^*).
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,0,-2,-4).

Cf. A018240, A090597, A336398.

a(n) = if (n%2, if ((n%4)==1, (2^(n-1)+2^((n-1)/2)-2)/3, (2^(n-1)+2^((n-1)/2))/3), (2^(n-1)+1)/3 + 2^(n/2-1));

a(n) = (2^(n-1)+1)/3 + 2^(n/2-1) if n is even; (2^(n-1)+2^((n-1)/2)-2)/3 if n is odd and n == 1 mod 4; (2^(n-1)+2^((n-1)/2))/3 if n is odd and n == 3 mod 4.
G.f.: x^2*(2 - 3*x^2 - 3*x^3 - 4*x^4)/(1 - x - 3*x^2 + x^3 + 2*x^5 + 4*x^6). - Jinyuan Wang, Jul 08 2020
From Wesley Ivan Hurt, Jul 17 2025: (Start)
a(n) = (2^(n+1)+2^(n/2)*(3+sqrt(2)+(-1)^n*(3-sqrt(2)))+4*((-1)^n+sin(3*n*Pi/2)))/12.
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 2*a(n-5) - 4*a(n-6). (End)

A352737 Number of oriented two-component rational links (or two-bridge links) with crossing number n (a chiral pair is counted as two distinct ones).

Original entry on oeis.org

2, 0, 4, 2, 10, 10, 30, 42, 102, 170, 374, 682, 1430, 2730, 5590, 10922, 22102, 43690, 87894, 174762, 350550, 699050, 1400150, 2796202, 5596502, 11184810, 22377814, 44739242, 89494870, 178956970, 357946710, 715827882, 1431721302, 2863311530, 5726754134, 11453246122

Offset: 2

Yuanan Diao, Mar 30 2022

The formula has been proved.

			If n=2 there are two rational links, namely, the Hopf link pair, one with positive crossings and the other with negative crossings. There are no two-component rational links with crossing number 3.

Yuanan Diao, Michael Lee Finney, Dawn Ray. The number of oriented rational links with a given deficiency number, Journal of Knot Theory and its Ramifications, Vol 30, Number 9, 2021. 2150065_1-20. See Theorem 4.3 and its proof.

Paolo Xausa, Table of n, a(n) for n = 2..1000
Yuanan Diao, Michael Lee Finney, and Dawn Ray, The number of oriented rational links with a given deficiency number, arXiv:2007.02819 [math.GT], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-4).

Cf. A336398, A336030.

LinearRecurrence[{1, 4, -2, -4}, {2, 0, 4, 2}, 50] (* Paolo Xausa, May 27 2024 *)

a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2); \\ Michel Marcus, Mar 31 2022

a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2).

G.f.: 2*x^2*(1 - x - 2*x^2 + x^3)/((1 + x)^(1 - 2*x)*(1 - 2*x^2)). - Stefano Spezia, Mar 31 2022

cp's OEIS Frontend

A018240 Number of rational knots (or two-bridge knots) with n crossings (up to mirroring).

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A090597 a(n) = - a(n-1) + 5(a(n-2) + a(n-3)) - 2(a(n-4) + a(n-5)) - 8(a(n-6) + a(n-7)).

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A329908 Number of oriented rational links with crossing number n.

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A352737 Number of oriented two-component rational links (or two-bridge links) with crossing number n (a chiral pair is counted as two distinct ones).

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