A329908 -id:A329908 - 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

A336398 Number of rational knots (or two-bridge knots) with n crossings (chiral pairs counted as distinct).

Original entry on oeis.org

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

Andrey Zabolotskiy, Jul 20 2020

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).

Cf. A018240, A090597, A329908.

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

(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).

A329909 Number of different four-digit proper fractions with anomalous cancellation property in base n.

Original entry on oeis.org

0, 0, 1, 15, 36, 135, 314, 588, 1173, 1851, 2806, 3433

Offset: 2

Xiaohan Zhang, Nov 23 2019

As in A329908, 0 digits or digits that appear different times in numerator and denominator are not cancelled.

Boas, R. P. "Anomalous Cancellation," Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979.

Eric Weisstein's World of Mathematics, Anomalous Cancellation

Cf. A094907 (two-digit), A329908 (three-digit).

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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A336398 Number of rational knots (or two-bridge knots) with n crossings (chiral pairs counted as distinct).

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A329909 Number of different four-digit proper fractions with anomalous cancellation property in base n.

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