A082761 Trinomial transform of the Fibonacci numbers (A000045).

1, 4, 20, 104, 544, 2848, 14912, 78080, 408832, 2140672, 11208704, 58689536, 307302400, 1609056256, 8425127936, 44114542592, 230986743808, 1209462292480, 6332826779648, 33159111507968, 173623361929216, 909103725543424

Offset

0,2

Comments

Hankel transform of sum{k=0..n, (-1)^k*C(2k,k)} (see A054108). - Paul Barry, Jan 13 2009

Hankel transform of A046748. - Paul Barry, Apr 14 2010

For positive n, a(n) equals the permanent of the (2n) X (2n) tridiagonal matrix with sqrt(2)'s along the three central diagonals. - John M. Campbell, Jul 12 2011

The limiting ratio a(n)/a(n-1) is 1 + phi^3. - Bob Selcoe, Mar 18 2014

Invert transform of A052984. Invert transform is A083066. Binomial transform of A033887. Binomial transform is A163073. - Michael Somos, May 26 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..150

László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.

Index entries for linear recurrences with constant coefficients, signature (6,-4).

Formula

a(n) = Sum[ Trinomial[n, k] Fibonacci[k+1], {k, 0, 2n} ] where Trinomial[n, k] = trinomial coefficients (A027907).

Third binomial transform of (1, 1, 5, 5, 25, 25, ....). a(n) = ((1+sqrt(5))(3+sqrt(5))^n-(1-sqrt(5))*(3-sqrt(5))^n)/(2*sqrt(5)). - Paul Barry, Jul 16 2003

G.f.: (1-2*x)/(1-6*x+4*x^2). a(n)= 6*a(n-1)-4*a(n-2). - R. J. Mathar, Nov 04 2008

a(n) = Sum_{k, 0<=k<=n} A147703(n,k)*3^k. - Philippe Deléham, Nov 14 2008

For n>=2: a(n) = 5*a(n-1) + sum_{i=1..n-2} a(i). - Bob Selcoe, Mar 18 2014

a(n) = a(-1-n) * 2^(2*n+1) for all n in Z. - Michael Somos, Mar 18 2014

a(n) = 2^n*Fibonacci(2*n+1), or 2^n*A001519(n+1). - Bob Selcoe, May 25 2014

a(n) - a(n-1) = A069429(n). a(n+1) * a(n-1) - a(n)^2 = 4^n. - Michael Somos, May 26 2014

G.f.: 1 / (1 - 4*x / (1 - x / (1 - x))). - Michael Somos, May 26 2014

Example

a(5) = 2848 = 5*(544)+4+20+104. - Bob Selcoe, Mar 18 2014
G.f. = 1 + 4*x + 20*x^2 + 104*x^3 + 544*x^4 + 2848*x^5 + 14912*x^6 + ...

Mathematica

a[ n_] := 2^n Fibonacci[ 2 n + 1]; (* Michael Somos, May 26 2014 *)
a[ n_] := If[ n < 0, SeriesCoefficient[ (2 - x) / (4 - 6 x + x^2), {x, 0, -1 - n}], SeriesCoefficient[ (1 - 2 x) / (1 - 6 x + 4 x^2), {x, 0, n}]]; (* Michael Somos, Oct 22 2017 *)
LinearRecurrence[{6, -4}, {1, 4}, 30] (* Harvey P. Dale, Jul 11 2014 *)

Prog

(PARI) a(n)=fibonacci(2*n+1)<<n \\ Charles R Greathouse IV, Jul 15 2011
(PARI) {a(n) = if( n<0, n = -1 - n; 2^(-1-2*n), 1) * polcoeff( (1 - 2*x) / (1 - 6*x + 4*x^2) + x * O(x^n), n)}; /* Michael Somos, Oct 22 2017 */
(Magma) [2^n * Fibonacci(2*n+1): n in [0..40]]; // Vincenzo Librandi, Jul 15 2011

Crossrefs

Cf. A001519, A033887, A046748, A052984, A069429, A083066, A163073.

Sequence in context: A226198 A155485 A155181 * A076035 A120978 A035028

Adjacent sequences: A082758 A082759 A082760 * A082762 A082763 A082764

Keyword

easy,nonn

Author

Emanuele Munarini, May 21 2003

Status

approved