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A049660 a(n) = F(6n)/8, where F=A000045 (the Fibonacci sequence). 29
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

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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 sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (18,-1).

FORMULA

G.f.: x/(1 - 18*x+ x^2).

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)

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 *)

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 */

CROSSREFS

Column m=6 of array A028412.

Cf. A134492. - Zerinvary Lajos, May 15 2009

Partial sums of A007805. - Wolfdieter Lang, Sep 01 2012

Cf. A001076, A001077, A014445, A014448, A015448, A099843, A155179.

Sequence in context: A158532 A214995 A171323 * A207697 A207593 A207512

Adjacent sequences:  A049657 A049658 A049659 * A049661 A049662 A049663

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

EXTENSIONS

More terms from James A. Sellers, Jan 20 2000

Chebyshev and other comments from Wolfdieter Lang, Nov 08 2002

More terms from Vladimir Joseph Stephan Orlovsky, Sep 11 2008

STATUS

approved

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Last modified September 29 04:37 EDT 2016. Contains 276609 sequences.