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%I #64 Dec 12 2023 09:23:09
%S 4,3,1,0,1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,1,0,
%T 1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,
%U 1,0,1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,1,0,1,3,4,3,1
%N Period 6: repeat [4, 3, 1, 0, 1, 3].
%C Third column of triangular array T defined in A131022.
%C a(n) = abs(A078070(n+1)).
%C Determinants of the spiral knots S(3,k,(1,1)). a(k+4) = det(S(3,k,(1,1))). These knots are also the torus knots T(3,k). - _Ryan Stees_, Dec 13 2014
%H A. Breiland, L. Oesper, and L. Taalman, <a href="http://educ.jmu.edu/~taalmala/breil_oesp_taal.pdf">p-Coloring classes of torus knots</a>, Online Missouri J. Math. Sci., 21 (2009), 120-126.
%H N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, <a href="http://projecteuclid.org/euclid.mjms/1312232716">Spiral knots</a>, Missouri J. of Math. Sci., 22 (2010).
%H M. DeLong, M. Russell, and J. Schrock, <a href="http://dx.doi.org/10.2140/involve.2015.8.361">Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m)</a>, Involve, Vol. 8 (2015), No. 3, 361-384.
%H Seong Ju Kim, R. Stees, and L. Taalman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Stees/stees4.html">Sequences of Spiral Knot Determinants</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
%H Ryan Stees, <a href="https://commons.lib.jmu.edu/honors201019/84">Sequences of Spiral Knot Determinants</a>, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,1).
%F a(1) = 4, a(2) = a(6) = 3, a(3) = a(5) = 1, a(4) = 0, a(6) = 1; for n > 6, a(n) = a(n-6).
%F G.f.: (4-5*x+3*x^2)/((1-x)*(1-x+x^2)).
%F a(n) = 2+cos(n*Pi/3)+sqrt(3)*sin(n*Pi/3) = 2+(-1)^((n-1)/3)+(-1)^((1-n)/3). - _Wesley Ivan Hurt_, Sep 11 2014
%F a(k+4) = det(S(3,k,(1,1))) = (b(k+4))^2, where b(5)=1, b(6)=sqrt(3), b(k)=sqrt(3)*b(k-1) - b(k-2) = b(6)*b(k-1) - b(k-2). - _Ryan Stees_, Dec 13 2014
%F a(n) = 2 + 2*cos(Pi/3*(n-1)) = 2 + A087204(n-1) for n >= 1. - _Werner Schulte_, Jul 18 2017 and _Peter Munn_, Apr 28 2022
%e For k=3, b(7)=sqrt(3)b(6)-b(5)=3-1=2, so det(S(3,3,(1,1)))=2^2=4.
%p A131027:=n->2+cos(n*Pi/3)+sqrt(3)*sin(n*Pi/3): seq(A131027(n), n=1..100); # _Wesley Ivan Hurt_, Sep 11 2014
%t Table[2 + Cos[n*Pi/3] + Sqrt[3]*Sin[n*Pi/3], {n, 30}] (* _Wesley Ivan Hurt_, Sep 11 2014 *)
%o (PARI) {m=105; for(n=1, m, r=(n-1)%6; print1(if(r==0, 4, if(r==1||r==5, 3, if(r==3, 0, 1))), ","))}
%o (Magma) m:=105; [ [4, 3, 1, 0, 1, 3][(n-1) mod 6 + 1]: n in [1..m] ];
%o (Sage) [(lucas_number2(n,2,1)-lucas_number2(n-1,1,1)) for n in range(4, 109)] # _Zerinvary Lajos_, Nov 10 2009
%Y Cf. A087204, A131022, A078070. Other columns of T are in A088911, A131026, A131028, A131029, A131030.
%K nonn,easy
%O 1,1
%A _Klaus Brockhaus_, following a suggestion of _Paul Curtz_, Jun 10 2007
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