OFFSET
0,1
COMMENTS
From L. Edson Jeffery, Apr 03 2011: (Start)
Let U be the unit-primitive matrix (see [Jeffery])
U = U_(9,1) =
(0 1 0 0)
(1 0 1 0)
(0 1 0 1)
(0 0 1 1).
Then a(n) = Trace(U^n). (End)
a(n)==1 (mod 3), a(3*n+1)==1 (mod 9). - Roman Witula, Sep 14 2012
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..3649
A. Akbary and Q. Wang, On some permutation polynomials over finite fields, International Journal of Mathematics and Mathematical Sciences, 2005:16 (2005) 2631-2640.
A. Akbary and Q. Wang, A generalized Lucas sequence and permutation binomials, Proceeding of the American Mathematical Society, 134 (1) (2006), 15-22, sequence a(n) with l=9.
David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, The Ramanujan Journal, Vol. 3, Issue 2, 1999, pp. 159-173.
David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, arXiv:0706.0356 [math.CA], 2007.
Russell A. Gordon, Lucas Type Sequences and Sums of Binomial Coefficients, Integers (2023) Vol 23, Art. No. A84. See p. 21.
L. E. Jeffery, Unit-primitive matrices
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.
Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141, Table 2 (k=4)
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1).
FORMULA
G.f.: ( 4-3*x-6*x^2+2*x^3 ) / ( (x-1)*(x^3+3*x^2-1) )
a(n) = 1+(2*cos(Pi/9))^n+(-2*sin(Pi/18))^n+(-2*cos(2*Pi/9))^n.
a(n) = 2^n*Sum_{k=1..4} cos((2*k-1)*Pi/9)^n. - L. Edson Jeffery, Apr 03 2011
a(n) = 1 + (-1)^n*A215664(n), which is compatible with the last two formulas above. - Roman Witula, Sep 14 2012
a(n) = 3*a(n-2) + a(n-3) - 3, with a(0)=4, a(1)=1, and a(2)=7. - Roman Witula, Sep 14 2012
EXAMPLE
We have a(0)+a(3)=a(1)+a(2)=8, a(3)+a(4)=a(2)+a(5)=23, and a(7)+a(8)=a(9)+a(3)=247. - Roman Witula, Sep 14 2012
MATHEMATICA
LinearRecurrence[{1, 3, -2, -1}, {4, 1, 7, 4}, 34] (* Jean-François Alcover, Sep 21 2017 *)
PROG
(PARI) Vec((4-3*x-6*x^2+2*x^3)/(1-x-3*x^2+2*x^3+x^4)+O(x^66)) /* Joerg Arndt, Apr 08 2011 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 18 2004
STATUS
approved