A130505 a(n) = 3*a(n-1) if n is odd, otherwise 6*a(n-1).

1, 3, 18, 54, 324, 972, 5832, 17496, 104976, 314928, 1889568, 5668704, 34012224, 102036672, 612220032, 1836660096, 11019960576, 33059881728, 198359290368, 595077871104, 3570467226624, 10711401679872, 64268410079232, 192805230237696

Offset: 0

Author

Gary W. Adamson, Jun 01 2007

Keywords

• nonn
• easy

Comments

Let x(n) be the upper left element of the n-th power of the matrix M = N*H, where H =[1,1; 1,-1] is a Hadamard matrix. Then x(1)=N, and x(2k+1) = N*x(2k), and x(2k) = 2N*x(2k-1), for all k>0. More precisely, it is easy to see (using H^2=2I) that even powers of M equal x(n)*I, where I = identity matrix, and odd powers of M are again multiples of the Hadamard matrix H, M^n = x(n)*H.

Here, N = 3, M = [3,3; 3,-3], M^4 = [324,0; 0,324] = a(4)*I with a(4) = 324, and H^3 = [54,54; 54,-54] = a(3)*H with a(3)=54.

This sequence is closely associated with (3+3*i)^n. (3+3*i)^n = a(n)*[s(n+2)+s(n)*i], where s(n) is of period 8 repeating(0,1,1,1,0,-1,-1,-1)= round(sin(Pi*n/4)). In general (a+a*i)^n = f(a,n)*[s(n+2)+s(n)*i], where f(a,n) = 2^floor(n/2)*a^n. - Gary Detlefs, Aug 04 2013.

Examples

			a(3) = 54 = 3 * a(2) = 3 * 18.
a(4) = 324 = 6 * a(3) = 6 * 54.
		

References

• K. J. Horadam, Hadamard Matrices and Their Applications, Princeton University Press, 2006.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,18).

Programs

Mathematica

CoefficientList[Series[(1 + 3*x)/(1 - 2*(3*x)^2), {x, 0, 50}], x] (* G. C. Greubel, Apr 17 2017 *)

PARI

a(n)=18^(n\2)*if(n%2,3,1) \\ Charles R Greathouse IV, Jan 10 2013

PARI

a(n)=([3,3; 3,-3]^n)[1,1] \\ Charles R Greathouse IV, Jan 10 2013

Formula

a(0) = 1; a(n), n>0 = 3*a(n-1) if n is odd. a(n), n even = 6 * a(n-1). Upper left term in M^n where M = the 2 X 2 matrix [3, 3; 3, -3].

a(n) = 18*a(n-2). - Charles R Greathouse IV, Jan 10 2013

a(n) = (3+3*i)^n/(s(n+2)+s(n)*i), where s(n)= round(sin(Pi*n/4)). - Gary Detlefs, Aug 04 2013

From Wolfdieter Lang, Sep 06 2013: (Start)

O.g.f.: (1+3*x)/(1-2*(3*x)^2) (see the Ch. R. Greathouse IV comment).

a(2*k) = 18^k, a(2*k+1) = 3*18^k, k>=0. (From the partial fraction decomposition of the o.g.f.) (End)

Sum_{n>=0} 1/a(n) = 24/17. - Amiram Eldar, Aug 27 2022

Extensions

Comment edited by Charles R Greathouse IV and M. F. Hasler, Jan 13 2013