A251610 Determinants of the spiral knots S(4,k,(1,1,1)).
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, 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, 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
Offset: 1
Examples
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.
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- A. Breiland, L. Oesper, and L. Taalman, p-Coloring classes of torus knots, Online Missouri J. Math. Sci., 21 (2009), 120-126.
- N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, Spiral knots, Missouri J. of Math. Sci., 22 (2010).
- M. DeLong, M. Russell, and J. Schrock, Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m), Involve, Vol. 8 (2015), No. 3, 361-384.
- Seong Ju Kim, R. Stees, L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
- Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
- Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).
Programs
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PARI
B=vector(166); B[1]=1; B[2]=s; \\ s := sqrt(2) for(n=3,#B,B[n]=s*B[n-1]-B[n-2]); B=substpol(B,s^2,2); A=vector(#B,n,n*B[n]^2); A=substpol(A,s^2,2) \\ Joerg Arndt, Dec 06 2014
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PARI
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
Formula
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).
From Colin Barker, Dec 06 2014: (Start)
b(k) = ((2-(-i)^k-i^k)*k)/2 where i=sqrt(-1).
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).
G.f.: x*(x^4+2*x^3-2*x^2+2*x+1) / ((x-1)^2*(x^2+1)^2).
(End)
Extensions
More terms from Joerg Arndt, Dec 06 2014
Comments