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

1,2

Comments

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

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).

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)

Example

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.

Prog

(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
(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

Crossrefs

Product of terms of A000027 and A007877.

Sequence in context: A304030 A338802 A242721 * A021703 A321209 A139823

Adjacent sequences: A251607 A251608 A251609 * A251611 A251612 A251613

Keyword

nonn,easy

Author

Ryan Stees, Dec 05 2014

Extensions

More terms from Joerg Arndt, Dec 06 2014

Status

approved