OFFSET
0,2
COMMENTS
We have a(n)=B(n;3), where B(n;d), n=1,2,..., d \in C, denote one of the quasi-Fibonacci numbers defined in the comments to A121449 and in the Witula-Slota-Warzynski paper. Its conjugate sequences A(n;3) and C(n;3) are discussed in A121458 and A215484 respectively. Similarly as in A121458 we deduce that each of the following elements a(3*n), a(3*n+1), a(3*n+2) is divided by 3*7^n for every n=0,1,... .
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (0, 21, 7).
FORMULA
a(n) = (1/7)*((c(1)-c(4))*(1+3*c(1))^n + (c(2)-c(1))*(1+3*c(2))^n + (c(4)-c(2))*(1+3*c(4))^n), where c(j):=2*cos(2*Pi*j/7) (for the proof see Witula-Slota-Warzynski paper).
G.f.: (3*x+6*x^2)/(1-21*x^2-7*x^3).
MATHEMATICA
LinearRecurrence[{0, 21, 7}, {0, 3, 6}, 50]
CoefficientList[Series[(3 x + 6 x^2)/(1 - 21 x^2 - 7 x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 18 2015 *)
PROG
(PARI) concat(0, Vec((3+6*x)/(1-21*x^2-7*x^3)+O(x^99))) \\ Charles R Greathouse IV, Oct 01 2012
(Magma) I:=[0, 3, 6]; [n le 3 select I[n] else 21*Self(n-2)+7*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roman Witula, Aug 13 2012
STATUS
approved