OFFSET
0,3
COMMENTS
We have a(n)=A(n;3), where A(n;d), n=1,2,..., d in C denote one of the three quasi-Fibonacci numbers defined in the comments to A121449 and in the Witula-Slota-Warzynski paper. The remaining two "conjugate" sequences B(n;3) and C(n;3) can be obtained from the following system of recurrence equations: A(0;3)=1, B(0;3)=C(0;3)=0, A(n+1;3)=A(n;3)+6*B(n;3)-3*C(n;3), B(n+1;3)=3*A(n;3)+B(n;3), C(n+1;3)=3*B(n;3)-2*C(n;3). These sequences are also connected by very intriguing convolution type relations (in some sense limiting in the nature) - see identities (3.53-55) and the identities (3.47-49) (the last ones for the value d=3) in the cited paper. We note that each of the three numbers a(3*n), a(3*n+1) and a(3*n+2) is divided by 7^n for every n=0,1,..., which follows easily from recurrence relation for the sequence a(n). - Roman Witula, Aug 11 2012
LINKS
Harvey P. Dale, 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(0)=a(1)=1, a(2)=19, a(n+1) = 21*a(n-1)+7*a(n-2) for n>=2.
a(n) = (1/7)*((s(2))^2*(1+3*c(1))^n + (s(4))^2*(1+3*c(2))^n + (s(1))^2*(1+3*c(4))^n), where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7) (for the proof see Witula-Slota-Warzynski paper). - Roman Witula, Aug 11 2012
MATHEMATICA
CoefficientList[Series[(1 + x - 2*x^2)/(1 - 21*x^2 - 7*x^3), {x, 0, 200}], x] (* Stefan Steinerberger, Sep 11 2006 *)
LinearRecurrence[{0, 21, 7}, {1, 1, 19}, 30] (* Harvey P. Dale, May 19 2012 *)
PROG
(PARI) Vec((1+x-2*x^2)/(1-21*x^2-7*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Sep 07 2006
EXTENSIONS
More terms from Stefan Steinerberger, Sep 11 2006
STATUS
approved