

A090597


a(n) =  a(n1) + 5(a(n2) + a(n3))  2(a(n4) + a(n5))  8(a(n6) + a(n7)).


6



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
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OFFSET

3,4


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(n2) 8s(n4): see A005418 (Jablan's observation) d(n) = d(n2) + 2d(n4): see A001045 (modified Jacobsthal sequence) l(n) = k(n1) + d(n): conjecture.
a(n) is the number of rational (2component) 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, 303315, 1987 (see Theorem 5, formulas for TL_n).
Index entries for linear recurrences with constant coefficients, signature (1,3,1,0,2,4).


FORMULA

a(n) = +a(n1) +3*a(n2) a(n3) 2*a(n5) 4*a(n6).  R. J. Mathar, Nov 23 2011
G.f.: x^4*(1+x^2+3*x^4+2*x^3) / ( (2*x1)*(1+x)*(2*x^21)*(1+x^2) ).  R. J. Mathar, Nov 23 2011
a(n) = (J(n3) + J((n3)/2))/2 if n is odd; (J(n3) + J(n/2))/2 if n is even, where J is the Jacobsthal number A001045.  David Scambler, Dec 12 2011


MATHEMATICA

f[x_] := (xx^32x^43x^5) / (1x3x^2+x^3+2x^5+4x^6); CoefficientList[ Series[ f[x], {x, 0, 29}], x] (* JeanFrançois 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 *)


PROG

(Haskell)
a090597 n = a090597_list !! (n3)
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


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.
Sequence in context: A276552 A213030 A303902 * A304887 A126073 A126592
Adjacent sequences: A090594 A090595 A090596 * A090598 A090599 A090600


KEYWORD

easy,nonn


AUTHOR

Thomas A. Gittings, Dec 11 2003


STATUS

approved



