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 A130505 a(n) = 3*a(n-1) if n is odd, otherwise 6*a(n-1). 2
 1, 3, 18, 54, 324, 972, 5832, 17496, 104976, 314928, 1889568, 5668704, 34012224, 102036672, 612220032, 1836660096, 11019960576, 33059881728, 198359290368, 595077871104, 3570467226624, 10711401679872, 64268410079232, 192805230237696 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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. 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). 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 EXAMPLE a(3) = 54 = 3 * a(2) = 3 * 18. a(4) = 324 = 6 * a(3) = 6 * 54. MATHEMATICA CoefficientList[Series[(1 + 3*x)/(1 - 2*(3*x)^2), {x, 0, 50}], x] (* G. C. Greubel, Apr 17 2017 *) PROG (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 CROSSREFS Sequence in context: A268484 A085789 A027334 * A222204 A027289 A061317 Adjacent sequences: A130502 A130503 A130504 * A130506 A130507 A130508 KEYWORD nonn,easy AUTHOR Gary W. Adamson, Jun 01 2007 EXTENSIONS Comment edited by Charles R Greathouse IV and M. F. Hasler, Jan 13 2013 STATUS approved

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Last modified January 30 19:55 EST 2023. Contains 359947 sequences. (Running on oeis4.)