login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A131027 Period 6: repeat [4, 3, 1, 0, 1, 3]. 12
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Third column of triangular array T defined in A131022.

a(n) = abs(A078070(n+1)).

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

LINKS

Table of n, a(n) for n=1..105.

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,-2,1).

FORMULA

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).

G.f.: (4-5*x+3*x^2)/((1-x)*(1-x+x^2)).

a(n) = 1/30*{-(n mod 6)-6*[(n+1) mod 6]-[(n+2) mod 6]+9*[(n+3) mod 6]+14*[(n+4) mod 6]+9*[(n+5) mod 6]}, with n>=0. - Paolo P. Lava, Jun 19 2007

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

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

a(n) = 2 + 2*cos(Pi/3*(n-1)) for n >= 1. - Werner Schulte, Jul 18 2017

EXAMPLE

For k=3, b(7)=sqrt(3)b(6)-b(5)=3-1=2, so det(S(3,3,(1,1)))=2^2=4.

MAPLE

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

MATHEMATICA

Table[2 + Cos[n*Pi/3] + Sqrt[3]*Sin[n*Pi/3], {n, 30}] (* Wesley Ivan Hurt, Sep 11 2014 *)

PROG

(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))), ", "))}

(MAGMA) m:=105; [ [4, 3, 1, 0, 1, 3][(n-1) mod 6 + 1]: n in [1..m] ];

(Sage) [(lucas_number2(n, 2, 1)-lucas_number2(n-1, 1, 1)) for n in range(4, 109)] # Zerinvary Lajos, Nov 10 2009

CROSSREFS

Cf. A131022, A078070. Other columns of T are in A088911, A131026, A131028, A131029, A131030.

Sequence in context: A285650 A144161 A054669 * A133475 A242106 A294885

Adjacent sequences:  A131024 A131025 A131026 * A131028 A131029 A131030

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 10 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 23 14:42 EDT 2021. Contains 348214 sequences. (Running on oeis4.)