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A099461 An Alexander sequence for the knot 9_48.

%I A099461 #23 Sep 08 2022 08:45:15
%S A099461 1,7,38,196,1001,5110,26093,133252,680510,3475339,17748434,90640627,
%T A099461 462898478,2364006148,12072895733,61655851222,314874250049,
%U A099461 1608051650884,8212262868470,41939735818687,214184746483778,1093833919809295,5586171115205846,28528378178106436,145693417671662033,744051127629095062,3799842775146922277,19405662567631938052,99104031922539424718
%N A099461 An Alexander sequence for the knot 9_48.
%C A099461 The denominator 1 -7*x +11*x^2 -7*x^3 +x^4 is a parameterization of the Alexander polynomial for the knot 9_48. 1/(1 -7*x +11*x^2 -7*x^3 +x^4) is the image of the g.f. of A099459 under the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)*A(x/(1+x^2)).
%H A099461 G. C. Greubel, <a href="/A099461/b099461.txt">Table of n, a(n) for n = 0..1000</a>
%H A099461 Dror Bar-Natan, <a href="http://katlas.org/wiki/9_48">9 48</a>, The Knot Atlas.
%H A099461 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-11,7,-1).
%F A099461 a(n) = A099460(n) - A099460(n-2).
%F A099461 G.f.: (1-x)*(1+x)*(1+x^2)/(1-7*x+11*x^2-7*x^3+x^4). - Corrected by _R. J. Mathar_, Nov 23 2012
%t A099461 LinearRecurrence[{7,-11,7,-1},{1,7,38,196,1001},40] (* _Harvey P. Dale_, Jun 18 2021 *)
%o A099461 (Magma) I:=[7,38,196,1001]; [1] cat [n le 4 select I[n] else 7*Self(n-1) - 11*Self(n-2) +7*Self(n-3) -Self(n-4): n in [1..41]]; // _G. C. Greubel_, Nov 18 2021
%o A099461 (Sage)
%o A099461 def A099461_list(prec):
%o A099461     P.<x> = PowerSeriesRing(ZZ, prec)
%o A099461     return P( (1-x)*(1+x)*(1+x^2)/(1-7*x+11*x^2-7*x^3+x^4) ).list()
%o A099461 A099461_list(40) # _G. C. Greubel_, Nov 18 2021
%Y A099461 Cf. A099459, A099460.
%K A099461 easy,nonn
%O A099461 0,2
%A A099461 _Paul Barry_, Oct 16 2004