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%I #47 Jul 20 2020 10:56:47
%S 0,1,1,3,3,8,12,27,45,96,176,363,693,1408,2752,5547,10965,22016,43776,
%T 87723,174933,350208,699392,1399467,2796885,5595136,11186176,22375083,
%U 44741973,89489408
%N 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)).
%C Arises from a conjecture about sequence of rational links with n crossings.
%C 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.
%C a(n) is the number of rational (2-component) links. - Slavik Jablan, Dec 26 2003
%C Also yields the number of meanders, reduced by symmetry, on an n X 3 rectangle (see A200893). - _Jon Wild_, Nov 25 2011
%H Reinhard Zumkeller, <a href="/A090597/b090597.txt">Table of n, a(n) for n = 3..1000</a>
%H C. Ernst and D. W. Sumners, <a href="https://doi.org/10.1017/S0305004100067323">The Growth of the Number of Prime Knots</a>, Math. Proc. Cambridge Philos. Soc. 102, 303-315, 1987 (see Theorem 5, formulas for TL_n).
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1,0,-2,-4).
%F 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
%F 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
%F 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
%t 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 *)
%t 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 *)
%t LinearRecurrence[{1,3,-1,0,-2,-4},{0,1,1,3,3,8},30] (* _Harvey P. Dale_, Nov 12 2013 *)
%o (Haskell)
%o a090597 n = a090597_list !! (n-3)
%o a090597_list = [0,1,1,3,3,8,12] ++ zipWith (-)
%o (drop 4 $ zipWith (-) (map (* 5) zs) (drop 2 a090597_list))
%o (zipWith (+) (drop 2 $ map (* 2) zs) (map (* 8) zs))
%o where zs = zipWith (+) a090597_list $ tail a090597_list
%o -- _Reinhard Zumkeller_, Nov 24 2011
%Y This is the difference between A005418 and A018240.
%Y Cf. A018240 = sequence of rational knots, A005418 = number of rational knots and links, A001045 = Jacobsthal sequence, A329908, A336398.
%Y Cf. A200893, and see the third column of the triangle read by rows there.
%K easy,nonn
%O 3,4
%A _Thomas A. Gittings_, Dec 11 2003
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