OFFSET
1,1
COMMENTS
From L. Edson Jeffery, Apr 05 2011: (Start)
Let U be the matrix (see [Jeffery])
U = U_(9,2) =
(0 0 1 0)
(0 1 0 1)
(1 0 1 1)
(0 1 1 1).
Then a(n) = Trace(U^n).
(End)
We note that all numbers of the form a(n)*3^(-floor((n+4)/3)) are integers. - Roman Witula, Sep 29 2012
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..200
L. Edson Jeffery, Danzer matrices
Index entries for linear recurrences with constant coefficients, signature (3,0,-3).
FORMULA
G.f.: x*(3 - 9*x^2)/(1 - 3*x + 3*x^3). The terms in parentheses in the definition are the roots of x^3-3*x^2+3. - Ralf Stephan, Apr 10 2004
a(n) = 3*(a(n-1) - a(n-3)) for n >= 4 - Roman Witula, Sep 29 2012
EXAMPLE
We have a(2)=3*a(1), a(4)/a(3) = a(6)/a(5) = a(7)/a(6) = 5/2, a(6)=6*a(4), a(7)=15*a(4), and (1 + c(1))^8 + (1 + c(2))^8 + (1 + c(4))^8 = 7*3^5. - Roman Witula, Sep 29 2012
MAPLE
Digits := 1000:q := seq(floor(evalf((1-2*cos(1/9*Pi))^n+(1+2*cos(2/9*Pi))^n+(1+2*cos(4/9*Pi))^n)), n=1..50);
MATHEMATICA
LinearRecurrence[{3, 0, -3}, {3, 9, 18}, 25] (* Georg Fischer Feb 02 2019 *)
PROG
(PARI) { default(realprecision, 200); for (n=1, 200, a=(1 - 2*cos(1/9*Pi))^n + (1 + 2*cos(2/9*Pi))^n + (1 + 2*cos(4/9*Pi))^n; write("b062882.txt", n, " ", round(a)) ) } \\ Harry J. Smith, Aug 12 2009
(PARI) Vec((3-9*x^2)/(1-3*x+3*x^3)+O(x^66)) /* Joerg Arndt, Apr 08 2011 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladeta Jovovic, Jun 27 2001
EXTENSIONS
More terms from Sascha Kurz, Mar 24 2002
Adapted formula, denominator of g.f. and modified g.f. (and offset) to accommodate added initial term a(0)=4. - L. Edson Jeffery, Apr 05 2011
a(0) = 4 removed, g.f. and programs adapted by Georg Fischer, Feb 02 2019
STATUS
approved