A121458 Expansion of (1+x-2*x^2)/(1-21*x^2-7*x^3).

1, 1, 19, 28, 406, 721, 8722, 17983, 188209, 438697, 4078270, 10530100, 88714549, 249679990, 1936716229, 5864281633, 42418800739, 136706927896, 931844786950, 3167777090989, 20525689021222, 73046232419419, 453213909082585

Offset: 0

Author

Philippe Deléham, Sep 07 2006

Keywords

• nonn
• easy

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).

Programs

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 *)

PARI

Vec((1+x-2*x^2)/(1-21*x^2-7*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

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

Extensions

More terms from Stefan Steinerberger, Sep 11 2006