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A090597
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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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3
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,4
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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
Comment from Slavik Jablan, Dec 26 2003: a(n) = number of rational (2-component) links.
Also yields the number of meanders, reduced by symmetry, on an n-by-3 rectangle (see A200893). - Jon Wild, Nov 25 2011
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
Index to sequences with linear recurrences with constant coefficients, signature (1,3,-1,0,-2,-4).
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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
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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] (* From 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 *)
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PROG
| (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
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CROSSREFS
| This is the difference between A005418 and A090596 (or A018240).
Cf. A018240 = sequence of rational knots, A005418 = number of rational knots and links, A001045 = Jacobsthal sequence, A090596.
Cf. A200893, and see the third column of the triangle read by rows there.
Sequence in context: A123315 A052407 A105039 * A126073 A126592 A055057
Adjacent sequences: A090594 A090595 A090596 * A090598 A090599 A090600
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KEYWORD
| easy,nonn
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AUTHOR
| Thomas A. Gittings (tomgittings(AT)aol.com), Dec 11 2003
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