A072985 Coefficient of the highest power of q in the expansion of nu(0)=1, nu(1)=b and for n >= 2, nu(n) = b*nu(n-1) + lambda*(n-1)_q*nu(n-2) with (b,lambda)=(2,3), where (n)_q = (1+q+...+q^(n-1)) and q is a root of unity.

1, 2, 7, 6, 21, 18, 63, 54, 189, 162, 567, 486, 1701, 1458, 5103, 4374, 15309, 13122, 45927, 39366, 137781, 118098, 413343, 354294, 1240029, 1062882, 3720087, 3188646, 11160261, 9565938, 33480783, 28697814, 100442349, 86093442

Offset

0,2

Comments

Instead of listing the coefficients of the highest power of q in each nu(n), if we list the coefficients of the smallest power of q (i.e., constant terms), we get a sequence of weighted Fibonacci numbers described by f(0)=1, f(1)=1, for n >= 2, f(n) = 2*f(n-1) + 3*f(n-2).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.

Index entries for linear recurrences with constant coefficients, signature (0,3).

Formula

For given b and lambda, the recurrence relation is given by; t(0)=1, t(1)=b, t(2) = b^2+lambda and for n >= 3, t(n) = lambda*t(n-2).

G.f.: (1 + 2*x + 4*x^2)/(1-3*x^2). - R. J. Mathar, Dec 05 2007

a(n) = 3*a(n-2) for n>2. - Ralf Stephan, Jul 19 2013

a(n) = (1/6)*(13 + (-1)^n)*3^floor(n/2) for n>0. - Ralf Stephan, Jul 19 2013

Example

nu(0) = 1;
nu(1) = 2;
nu(2) = 7;
nu(3) = 20 + 6q;
nu(4) = 61 + 33q + 21q^2;
nu(5) = 182 + 144q + 120q^2 + 78q^3 + 18q^4;
nu(6) = 547 + 570q + 585q^2 + 501q^3 + 381q^4 + 162q^5 + 63q^6; ...
The coefficients of the highest power of q give this sequence.

Mathematica

CoefficientList[Series[-(1 + 2 x + 4 x^2) / (-1 + 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 20 2013 *)
Join[{1}, LinearRecurrence[{0, 3}, {2, 7}, 33]] (* Jean-François Alcover, Sep 23 2017 *)

Prog

(Magma) [1] cat [(1/6)*(13+(-1)^n)*3^Floor(n/2): n in [1..40]]; // Vincenzo Librandi, Jul 20 2013
(PARI) x='x+O('x^30); Vec((1+2*x+4*x^2)/(1-3*x^2)) \\ G. C. Greubel, May 26 2018

Crossrefs

Cf. A014983.

Sequence in context: A338041 A365007 A082017 * A371597 A082187 A211368

Adjacent sequences: A072982 A072983 A072984 * A072986 A072987 A072988

Keyword

nonn,easy

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Extensions

More terms from R. J. Mathar, Dec 05 2007

Status

approved