A074324 a(2n+1) = 3^n, a(2n) = 4*3^(n-1) except for a(0) = 1.

1, 1, 4, 3, 12, 9, 36, 27, 108, 81, 324, 243, 972, 729, 2916, 2187, 8748, 6561, 26244, 19683, 78732, 59049, 236196, 177147, 708588, 531441, 2125764, 1594323, 6377292, 4782969, 19131876, 14348907, 57395628, 43046721, 172186884, 129140163

Offset

0,3

Comments

Also: 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)=(1,3), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity.

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 weighted Fibonacci numbers described by f(0)=1, f(1)=1, for n>=2, f(n)=f(n-1)+3f(n-2).

Sequences A162766, A166552 are essentially the same. - M. F. Hasler, Dec 03 2014

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+x+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)*(7+(-1)^n)*3^floor(n/2) for n>0. - Ralf Stephan, Jul 19 2013

Example

nu(0)=1;
nu(1)=1;
nu(2)=4;
nu(3)=7+3q;
nu(4)=19+15q+12q^2;
nu(5)=40+45q+42q^2+30q^3+9q^4;
nu(6)=97+147q+180q^2+168q^3+147q^4+81q^5+36q^6;
by listing the coefficients of the highest power in each nu(n), we get, 1,1,4,3,12,9,36,....

Mathematica

CoefficientList[Series[-(1 + x + x^2) / (-1 + 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 20 2013 *)
LinearRecurrence[{0, 3}, {1, 1, 4}, 40] (* Harvey P. Dale, Mar 13 2016 *)

Prog

(Magma) [1] cat [(1/6)*(7+(-1)^n)*3^Floor(n/2):n in [1..40]]; // Vincenzo Librandi, Jul 20 2013
(PARI) a(n)=3^(n\2)\(3/4)^!bittest(n, 0) \\ M. F. Hasler, Dec 03 2014

Crossrefs

Cf. A006130.

Sequence in context: A055527 A055523 A168430 * A162766 A166552 A122804

Adjacent sequences: A074321 A074322 A074323 * A074325 A074326 A074327

Keyword

nonn,easy

Author

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

Extensions

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

Simpler definition from M. F. Hasler, Dec 03 2014

Status

approved