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 A049660 a(n) = Fibonacci(6*n)/8. 32
 0, 1, 18, 323, 5796, 104005, 1866294, 33489287, 600940872, 10783446409, 193501094490, 3472236254411, 62306751484908, 1118049290473933, 20062580477045886, 360008399296352015, 6460088606857290384, 115921586524134874897, 2080128468827570457762 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS For n>=2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 18's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011 For n>=2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,17}. - Milan Janjic, Jan 25 2015 10*a(n)^2 = Tri(4)*S(n-1, 18)^2 is the triangular number Tri((T(n, 9) - 1)/2), with Tri, S and T given in A000217, A049310 and A053120. This is instance k = 4 of the k-family of identities given in a comment on A001109. - Wolfdieter Lang, Feb 01 2016 LINKS Indranil Ghosh, Table of n, a(n) for n = 0..796 (terms 0..200 from Vincenzo Librandi) R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014). Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (18,-1). FORMULA G.f.: x/(1 - 18*x+ x^2). a(n) = A134492(n)/8. a(n) ~ (1/40)*sqrt(5)*(sqrt(5) + 2)^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002 For all elements x of the sequence, 80*x^2 + 1 is a square. Lim_{n->inf.} a(n)/a(n-1) = 8*phi + 5 = 9 + 4*sqrt(5). - Gregory V. Richardson, Oct 14 2002 a(n) = S(n-1, 18) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the second kind. S(-1, x) := 0. See A049310. a(n) = (((9+4*sqrt(5))^n - (9-4*sqrt(5))^n))/(8*sqrt(5)). a(n) = sqrt((A023039(n)^2 - 1)/80) (cf. Richardson comment). a(n) = 18*a(n-1) - a(n-2). - Gregory V. Richardson, Oct 14 2002 a(n) = A001076(2n)/4. a(n) = 17*(a(n-1) + a(n-2)) - a(n-3) = 19*(a(n-1) - a(n-2)) + a(n-3). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 26 2007 a(n+1) = Sum_{k=0..n} A101950(n,k)*17^k. - Philippe Deléham, Feb 10 2012 Product_{n>=1} (1 + 1/a(n)) = 1/2*(2 + sqrt(5)). - Peter Bala, Dec 23 2012 Product_{n>=2} (1 - 1/a(n)) = 2/9*(2 + sqrt(5)). - Peter Bala, Dec 23 2012 a(n) = 1/32*(F(6*n + 3) - F(6*n - 3)). Sum_{n>=1} 1/(4*a(n) + 1/(4*a(n))) = 1/4. Compare with A001906 and A049670. - Peter Bala, Nov 29 2013 From Peter Bala, Apr 02 2015: (Start) Sum_{n >= 1} a(n)*x^(2*n) = -G(x)*G(-x), where G(x) = Sum_{n >= 1} A001076(n)*x^n. 1 + 4*Sum_{n >= 1} a(n)*x^(2*n) = (1 + F(x))*(1 + F(-x)) = (1 + 2*x*G(x))*(1 - 2*x*G(-x)), where F(x) = Sum_{n >= 1} Fibonacci(3*n + 3)*x^n. 1 + 7*Sum_{n >= 1} a(n)*x^(2*n) = (1 + G(x))*(1 + G(-x)) = (1 + 7*G(x))*(1 + 7*G(-x)). 1 + 12*Sum_{n >= 1} a(n)*x^(2*n) = (1 + 2*G(x))*(1 + 2*G(-x)) = (1 + 6*G(x))*(1 + 6*G(-x)) = (1 + A(x))*(1 + A(-x)), where A(x) = Sum_{n >= 1} Fibonacci(3*n)*x^n is the o.g.f for A014445. 1 + 15*Sum_{n >= 1} a(n)*x^(2*n) = (1 + 5*G(x))*(1 + 5*G(-x)) = (1 + 3*G(x))*(1 + 3*G(-x)) = H(x)*H(-x), where H(x) = Sum_{n >= 0} A155179(n)*x^n. 1 + 16*Sum_{n >= 1} a(n)*x^(2*n) = (1 + 4*G(x))*(1 + 4*G(-x)) = (1 + 2* Sum_{n >= 1} Fibonacci(3*n - 1)*x^n)*(1 + 2* Sum_{n >= 1} Fibonacci(3*n - 1)*(-x)^n) = (1 + 2* Sum_{n >= 1} Fibonacci(3*n + 1)*x^n)*(1 + 2* Sum_{n >= 1} Fibonacci(3*n + 1)*(-x)^n). 1 + 20*Sum_{n >= 1} a(n)*x^(2*n) = (1 + Sum_{n >= 1} Lucas(3*n)*x^n)*(1 + Sum_{n >= 1} Lucas(3*n)*(-x)^n). 1 - 5*Sum_{n >= 1} a(n)*x^(2*n) = (1 + Sum_{n >= 1} A001077(n+1)*x^n)*(1 + Sum_{n >= 1} A001077(n+1)*(-x)^n). 1 - 9*Sum_{n >= 1} a(n)*x^(2*n) = (1 - G(x))*(1 - G(-x)) = (1 + 9*G(x))*(1 + 9*G(-x)). 1 - 16*Sum_{n >= 1} a(n)*x^(2*n) = (1 + 2*Sum_{n >= 1} A099843(n)*x^n)*(1 + 2*Sum_{n >= 1} A099843(n)*(-x)^n). 1 - 20*Sum_{n >= 1} a(n)*x^(2*n) = (1 - 2*G(x))*(1 - 2*G(-x)) = (1 + 10*G(x))*(1 + 10*G(-x)). (End) EXAMPLE a(3) = F(6 * 3) / 8 = F(18) / 8 = 2584 / 8 = 323. - Indranil Ghosh, Feb 06 2017 MAPLE with (combinat):seq(fibonacci(2*n, 4)/4, n=0..16); # Zerinvary Lajos, Apr 20 2008 MATHEMATICA Fibonacci[6*Range[0, 20]]/8 (* Harvey P. Dale, Nov 23 2011 *) LinearRecurrence[{18, -1}, {0, 1}, 30] (* G. C. Greubel, Dec 02 2017 *) PROG (MuPAD) numlib::fibonacci(6*n)/8 \$ n = 0..25; // Zerinvary Lajos, May 09 2008 (Sage) [lucas_number1(n, 18, 1) for n in xrange(0, 20)] # Zerinvary Lajos, Jun 25 2008 (Sage) [fibonacci(6*n)/8 for n in xrange(0, 17)] # Zerinvary Lajos, May 15 2009 (PARI) a(n)=fibonacci(6*n)/8  \\ Charles R Greathouse IV, Apr 17 2012 (MAGMA) [Fibonacci(6*n)/8: n in [0..30]]; // G. C. Greubel, Dec 02 2017 CROSSREFS Column m=6 of array A028412. Partial sums of A007805. Cf. A000045, A001076, A001077, A014445, A014448, A015448, A099843, A134492, A155179. Sequence in context: A158532 A214995 A171323 * A207697 A207593 A207512 Adjacent sequences:  A049657 A049658 A049659 * A049661 A049662 A049663 KEYWORD nonn,easy AUTHOR EXTENSIONS Chebyshev and other comments from Wolfdieter Lang, Nov 08 2002 STATUS approved

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Last modified July 23 00:51 EDT 2018. Contains 312919 sequences. (Running on oeis4.)