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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
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 (list; graph; refs; listen; history; text; internal format)
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: A168205 A265495 A082017 * A082187 A211368 A225101

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

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Last modified July 15 00:36 EDT 2020. Contains 335762 sequences. (Running on oeis4.)