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)).

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

Author

Thomas A. Gittings, Dec 11 2003

Keywords

• easy
• nonn

Comments

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

Links

• 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).

Crossrefs

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.

Programs

Haskell

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

Mathematica

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 *)

Formula

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