Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #21 May 09 2024 09:07:48
%S 1,5,18,60,197,650,2153,7140,23682,78545,260498,863945,2865282,
%T 9502740,31515953,104523050,346651997,1149675660,3812913618,
%U 12645575405,41939208002,139091904605,461300030418,1529907284460,5073956524397
%N An Alexander sequence for the knot 7_6.
%C The denominator is a parameterization of the Alexander polynomial for the knot 7_6. The g.f. is the image of the g.f. of A030191 under the modified Chebyshev transform A(x)->(1/(1+x^2)^2)A(x/(1+x^2))
%H Vincenzo Librandi, <a href="/A099449/b099449.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="http://www.math.toronto.edu/~drorbn/KAtlas/Knots/">The Rolfsen Knot Table</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,5,-1).
%F G.f.: -(x-1)*(x+1)*(x^2+1) / (x^4-5*x^3+7*x^2-5*x+1). - _Colin Barker_, Feb 10 2014
%F a(n) = A099448(n) - A099448(n-2).
%F a(n) = 5*a(n-1)-7*a(n-2)+5*a(n-3)-a(n-4) for n>4. - _Colin Barker_, Feb 10 2014
%t CoefficientList[Series[(1 - x) (x + 1) (x^2 + 1)/(x^4 -5 x^3 + 7 x^2 - 5 x + 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)
%t LinearRecurrence[{5,-7,5,-1},{1,5,18,60,197},30] (* _Harvey P. Dale_, Oct 06 2015 *)
%o (PARI) Vec(-(x-1)*(x+1)*(x^2+1)/(x^4-5*x^3+7*x^2-5*x+1) + O(x^100)) \\ _Colin Barker_, Feb 10 2014
%o (Magma) I:=[1,5,18,60,197,650,2153,7140]; [n le 8 select I[n] else 5*Self(n-1)-7*Self(n-2)+5*Self(n-3)-Self(n-4) : n in [1..30]]; // _Vincenzo Librandi_, Feb 12 2014
%Y Cf. A030191, A099448.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Oct 16 2004
%E g.f. corrected by _Colin Barker_, Feb 10 2014