|
| |
|
|
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, and 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)) = 2 + A087204(n-1) for n >= 1. - Werner Schulte, Jul 18 2017 and Peter Munn, Apr 28 2022
|
|
|
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. A087204, 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
|
| |
|
|
