OFFSET
0,3
COMMENTS
From Roman Witula, Aug 07 2012: (Start)
In the cited Witula-Slota-Warzynski paper three so-called quasi-Fibonacci numbers A(n;d), B(n;d) and C(n;d), where n = 0,1,...,d \in C are discussed. These numbers are created by each of the following relations:
(1+d*c(j))^n = A(n;d) + B(n;d)*c(j) + C(n;d)*c(2*j), for every j=1,2,4, where c(j):=2*cos(2*Pi*j/7).
In fact all these "numbers" are integer polynomials of the argument d.
In the sequel for d=-1 we obtain A(n;-1)=a(n), B(n+1;-1)=-A085810(n).
We note that the elements of the sequences A(n;-1), B(n;-1), and C(n;-1) satisfy the following system of recurrence equations:
A(0;-1)=1, B(0;-1)=C(0;-1)=0,
A(n+1;-1)=A(n;-1)-2*B(n;-1)+C(n;-1),
B(n+1;-1)=-A(n;-1)+B(n;-1), C(n+1;-1)=-B(n;-1)+2*C(n;-1).
It is proved that binomial transforms of the sequences: A(n;1), B(n;1), and C(n;1) are equal to the following sequences:
A(n;1)*(A(n;-1)-C(n;-1))-B(n;1)*(B(n;-1)+C(n;-1))+C(n;1)*B(n;-1), -A(n;1)*C(n;-1)+B(n;1)*(A(n;-1)-C(n;-1))-C(n;1)*(B(n;-1)-C(n;-1)), and
A(n;1)*(B(n;-1)-C(n;-1))-B(n;1)*B(n;-1)+C(n;1)*(A(n;-1)-B(n;-1)+C(n;-1)), respectively, whereas we have
A(n;-1) = Sum_{k=0..n} binomial(n,k)*(A(k;1)*A(n-k;1)-A(k;1)*B(n-k;1)-B(k;1)*C(n-k;1)-A(n-k;1)*C(k;1)+2*B(n-k;1)*C(k;1)-C(k;1)*C(n-k;1)),
B(n;-1) = Sum_{k=0..n} binomial(n,k)*(-A(k;1)*B(n-k;1)+A(k;1)*C(n-k;1)+B(k;1)*B(n-k;1)-A(n-k;1)*C(k;1)+B(n-k;1)*C(k;1)-C(k;1)*C(n-k;1)), and
C(n;-1) = Sum_{k=0..n} binomial(n,k)*(-A(k;1)*B(n-k;1)+A(n-k;1)*B(k;1)+B(k;1)*B(n-k;1)-B(k;1)*C(n-k;1)-A(n-k;1)*C(k;1)) (see identities (3.50-52) and (3.61-63) in the Witula-Slota-Warzynski paper).
(End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
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 (4,-3,-1).
FORMULA
a(0)=a(1)=1, a(2)=3, a(n+1) = 4*a(n) - 3*a(n-1) - a(n-2) for n>=2.
7*a(n) = (2-c(4))*(1-c(1))^n + (2-c(1))*(1-c(2))^n + (2-c(2))*(1-c(4))^n = (s(2))^2*(1-c(1))^n + (s(4))^2*(1-c(2))^n + (s(1))^2*(1-c(4))^n, where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7) -- it is the special case, for d=-1, of the Binet's formula for the respective quasi-Fibonacci number A(n;d) discussed in the Witula-Slota-Warzynski paper. - Roman Witula, Aug 07 2012
MATHEMATICA
CoefficientList[Series[(1 - 3*x + 2*x^2)/(1-4*x + 3*x^2 + x^3), {x, 0, 200}], x] (* Stefan Steinerberger, Sep 11 2006 *)
LinearRecurrence[{4, -3, -1}, {1, 1, 3}, 50] (* Roman Witula, Aug 07 2012 *)
PROG
(Magma) I:=[1, 1, 3]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015
(PARI) x='x+O('x^30); Vec((1-3*x+2*x^2)/(1-4*x+3*x^2+x^3)) \\ G. C. Greubel, Apr 19 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Sep 06 2006
EXTENSIONS
More terms from Stefan Steinerberger, Sep 11 2006
a(27)-a(29) from Vincenzo Librandi, Sep 18 2015
STATUS
approved