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 A099445 An Alexander sequence for the Miller Institute knot. 2
 1, 3, 6, 12, 25, 54, 117, 252, 542, 1167, 2514, 5415, 11662, 25116, 54093, 116502, 250913, 540396, 1163862, 2506635, 5398594, 11627067, 25041462, 53932332, 116155217, 250165974, 538787805, 1160398812, 2499175726, 5382528183 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The denominator is a parameterization of the Alexander polynomial for the knot 6_2 (Miller Institute knot). The g.f. is the image of that of Fib(2n+2) under the modified Chebyshev transform A(x)->(1/(1+x^2)^2)A(x/(1+x^2)). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Dror Bar-Natan, The Rolfsen Knot Table Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-1). FORMULA G.f.: -(x-1)*(x+1)*(x^2+1) / (x^4-3*x^3+3*x^2-3*x+1). - Colin Barker, Feb 10 2014 a(n) = A099444(n)-A099444(n-2). a(n) = 3*a(n-1)-3*a(n-2)+3*a(n-3)-a(n-4) for n>4. - Colin Barker, Feb 10 2014 MATHEMATICA CoefficientList[Series[(1 - x) (x + 1) (x^2 + 1)/(x^4 - 3 x^3 + 3 x^2 - 3 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *) LinearRecurrence[{3, -3, 3, -1}, {1, 3, 6, 12, 25}, 30] (* Harvey P. Dale, Jun 24 2018 *) PROG (PARI) Vec(-(x-1)*(x+1)*(x^2+1)/(x^4-3*x^3+3*x^2-3*x+1) + O(x^100)) \\ Colin Barker, Feb 10 2014 (Magma) I:=[1, 3, 6, 12, 25, 54, 117, 252]; [n le 8 select I[n] else 3*Self(n-1)-3*Self(n-2)+3*Self(n-3)-Self(n-4) : n in [1..50]]; // Vincenzo Librandi, Feb 12 2014 CROSSREFS Cf. A001906. Sequence in context: A243728 A243721 A280109 * A004067 A278821 A274476 Adjacent sequences: A099442 A099443 A099444 * A099446 A099447 A099448 KEYWORD easy,nonn AUTHOR Paul Barry, Oct 16 2004 EXTENSIONS g.f. corrected by Colin Barker, Feb 10 2014 STATUS approved

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Last modified February 21 22:50 EST 2024. Contains 370239 sequences. (Running on oeis4.)