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A099445 An Alexander sequence for the Miller Institute knot.

%I A099445 #23 Apr 06 2025 07:39:00
%S A099445 1,3,6,12,25,54,117,252,542,1167,2514,5415,11662,25116,54093,116502,
%T A099445 250913,540396,1163862,2506635,5398594,11627067,25041462,53932332,
%U A099445 116155217,250165974,538787805,1160398812,2499175726,5382528183
%N A099445 An Alexander sequence for the Miller Institute knot.
%C A099445 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)).
%H A099445 Vincenzo Librandi, <a href="/A099445/b099445.txt">Table of n, a(n) for n = 0..1000</a>
%H A099445 Dror Bar-Natan, <a href="https://katlas.org/wiki/The_Rolfsen_Knot_Table">The Rolfsen Knot Table</a>
%H A099445 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3,-1).
%F A099445 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
%F A099445 a(n) = A099444(n)-A099444(n-2).
%F A099445 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
%t A099445 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 *)
%t A099445 LinearRecurrence[{3,-3,3,-1},{1,3,6,12,25},30] (* _Harvey P. Dale_, Jun 24 2018 *)
%o A099445 (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
%o A099445 (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
%Y A099445 Cf. A001906.
%K A099445 easy,nonn
%O A099445 0,2
%A A099445 _Paul Barry_, Oct 16 2004
%E A099445 G.f. corrected by _Colin Barker_, Feb 10 2014