A155179 -id:A155179 - cp's OEIS frontend

A049660 a(n) = Fibonacci(6*n)/8.

0, 1, 18, 323, 5796, 104005, 1866294, 33489287, 600940872, 10783446409, 193501094490, 3472236254411, 62306751484908, 1118049290473933, 20062580477045886, 360008399296352015, 6460088606857290384, 115921586524134874897, 2080128468827570457762

Offset: 0

Clark Kimberling

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
Possible solutions for y in Pell equation x^2 - 80*y^2 = 1. The values for x are given in A023039. - Herbert Kociemba, Jun 05 2022

			a(3) = F(6 * 3) / 8 = F(18) / 8 = 2584 / 8 = 323. - _Indranil Ghosh_, Feb 06 2017

Indranil Ghosh, Table of n, a(n) for n = 0..796 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
R. Flórez, R. A. Higuita, and 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).

Column m=6 of array A028412.
Partial sums of A007805.
Cf. A000045, A001076, A001077, A014445, A014448, A015448, A099843, A134492, A155179.

[Fibonacci(6*n)/8: n in [0..30]]; // G. C. Greubel, Dec 02 2017

with (combinat):seq(fibonacci(2*n,4)/4, n=0..16); # Zerinvary Lajos, Apr 20 2008

Fibonacci[6*Range[0,20]]/8 (* Harvey P. Dale, Nov 23 2011 *)
LinearRecurrence[{18,-1}, {0,1}, 30] (* G. C. Greubel, Dec 02 2017 *)
Table[ChebyshevU[-1 + n, 9], {n, 0, 18}] (* Herbert Kociemba, Jun 05 2022 *)

numlib::fibonacci(6*n)/8 $ n = 0..25; // Zerinvary Lajos, May 09 2008

a(n)=fibonacci(6*n)/8  \\ Charles R Greathouse IV, Apr 17 2012

[lucas_number1(n,18,1) for n in range(0,20)] # Zerinvary Lajos, Jun 25 2008

[fibonacci(6*n)/8 for n in range(0, 17)] # Zerinvary Lajos, May 15 2009

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 terms k of the sequence, 80*k^2 + 1 is a square. Limit_{n->oo} 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, 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)

Chebyshev and other comments from Wolfdieter Lang, Nov 08 2002

A155161 A Fibonacci convolution triangle: Riordan array (1, x/(1 - x - x^2)). Triangle T(n,k), 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 3, 1, 0, 5, 10, 9, 4, 1, 0, 8, 20, 22, 14, 5, 1, 0, 13, 38, 51, 40, 20, 6, 1, 0, 21, 71, 111, 105, 65, 27, 7, 1, 0, 34, 130, 233, 256, 190, 98, 35, 8, 1, 0, 55, 235, 474, 594, 511, 315, 140, 44, 9, 1, 0, 89, 420, 942, 1324, 1295, 924, 490, 192, 54, 10, 1

Offset: 0

Philippe Deléham, Jan 21 2009

			Triangle begins:
[0] 1;
[1] 0,  1;
[2] 0,  1,   1;
[3] 0,  2,   2,   1;
[4] 0,  3,   5,   3,   1;
[5] 0,  5,  10,   9,   4,   1;
[6] 0,  8,  20,  22,  14,   5,  1;
[7] 0, 13,  38,  51,  40,  20,  6,  1;
[8] 0, 21,  71, 111, 105,  65, 27,  7, 1;
[9] 0, 34, 130, 233, 256, 190, 98, 35, 8, 1.

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

Row sums are in A215928.
Central terms: T(2*n,n) = A213684(n) for n > 0.
Cf. A000045, A037027, A122542, A059283, A321620.

a155161 n k = a155161_tabl !! n !! k
a155161_row n = a155161_tabl !! n
a155161_tabl = [1] : [0,1] : f [0] [0,1] where
   f us vs = ws : f vs ws where
     ws = zipWith (+) (us ++ [0,0]) $ zipWith (+) ([0] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Apr 17 2013

T := (n, k) -> binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..11); # Peter Luschny, May 23 2021
# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-fibonacci(n)); # Peter Luschny, Oct 07 2022

CoefficientList[#, y]& /@ CoefficientList[(1-x-x^2)/(1-x-x^2-x*y)+O[x]^12, x] // Flatten (* Jean-François Alcover, Mar 01 2019 *)
(* Generates the triangle without the leading '1' (rows are rearranged). *)
(* Function RiordanSquare defined in A321620. *)
RiordanSquare[x/(1 - x - x^2), 11] // Flatten  (* Peter Luschny, Feb 27 2021 *)

M(n,k):=pochhammer(n,k)/k!;
create_list(sum(M(k,i)*binomial(i,n-i-k),i,0,n-k),n,0,8,k,0,n); /* Emanuele Munarini, Mar 15 2011 */

T(n, k) given by [0,1,1,-1,0,0,0,...] DELTA [1,0,0,0,...] where DELTA is the operator defined in A084938.
a(n,k) = Sum_{i=0..n-k} M(k,i)*binomial(i,n-i-k), where M(n,k) = n(n+1)(n+2)...(n+k-1)/k!. - Emanuele Munarini, Mar 15 2011
Recurrence: a(n+2,k+1) = a(n+1,k+1) + a(n+1,k) + a(n,k+1). - Emanuele Munarini, Mar 15 2011
G.f.: (1-x-x^2)/(1-x-x^2-x*y). - Philippe Deléham, Feb 08 2012
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000129(n) (n > 0), A052991(n), A155179(n), A155181(n), A155195(n), A155196(n), A155197(n), A155198(n), A155199(n) for x = 0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Feb 08 2012
T(n, k) = binomial(n-1, k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], -4). - Peter Luschny, May 23 2021

cp's OEIS Frontend

A049660 a(n) = Fibonacci(6*n)/8.

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