A251610 - OEIS

%I #49 Nov 22 2020 12:20:19

%S 1,4,3,0,5,12,7,0,9,20,11,0,13,28,15,0,17,36,19,0,21,44,23,0,25,52,27,

%T 0,29,60,31,0,33,68,35,0,37,76,39,0,41,84,43,0,45,92,47,0,49,100,51,0,

%U 53,108,55,0,57,116,59,0,61,124,63,0,65,132,67,0,69,140,71,0,73,148,75,0,77,156,79

%N Determinants of the spiral knots S(4,k,(1,1,1)).

%C a(k) = det(S(4,k,(1,1,1))). These knots are also the torus knots T(4,k).

%H Colin Barker, <a href="/A251610/b251610.txt">Table of n, a(n) for n = 1..1000</a>

%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, 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_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,4,-3,2,-1).

%F a(k) = det(S(4,k,(1,1,1))) = k*(b(k))^2, where b(1)=1, b(2)=sqrt(2), b(k)=sqrt(2)*b(k-1) - b(k-2) = b(2)*b(k-1) - b(k-2).

%F From _Colin Barker_, Dec 06 2014: (Start)

%F b(k) = ((2-(-i)^k-i^k)*k)/2 where i=sqrt(-1).

%F b(k) = 2*b(k-1)-3*b(k-2)+4*b(k-3)-3*b(k-4)+2*b(k-5)-b(k-6).

%F G.f.: x*(x^4+2*x^3-2*x^2+2*x+1) / ((x-1)^2*(x^2+1)^2).

%F (End)

%e For k=3, b(3)=sqrt(2)b(2)-b(1)=2-1=1, so det(S(4,3,(1,1,1)))=3*1^2=3.

%o (PARI)

%o B=vector(166); B[1]=1; B[2]=s;  \\ s := sqrt(2)

%o for(n=3,#B,B[n]=s*B[n-1]-B[n-2]);

%o B=substpol(B,s^2,2);

%o A=vector(#B,n,n*B[n]^2);

%o A=substpol(A,s^2,2)

%o \\ _Joerg Arndt_, Dec 06 2014

%o (PARI)

%o Vec(x*(x^4+2*x^3-2*x^2+2*x+1) / ((x-1)^2*(x^2+1)^2) + O(x^100)) \\ _Colin Barker_, Dec 07 2014

%Y Product of terms of A000027 and A007877.

%K nonn,easy

%O 1,2

%A _Ryan Stees_, Dec 05 2014

%E More terms from _Joerg Arndt_, Dec 06 2014