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, 4760128905543680, 24924358531088384, 130505635564355584

Offset: 0

Author

Emanuele Munarini, May 21 2003

Keywords

• easy
• nonn

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 is: Lim_{n -> oo} a(n)/a(n-1) = 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

Examples

			a(5) = 2848 = 5*(544) + 4 + 20 + 104.
G.f. = 1 + 4*x + 20*x^2 + 104*x^3 + 544*x^4 + 2848*x^5 + 14912*x^6 + ...
		

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).

Crossrefs

Cf. A000045, A001519, A027907, A033887, A046748, A052984, A054108, A069429, A083066, A147703, A163073.

Programs

Magma

[2^n * Fibonacci(2*n+1): n in [0..40]]; // Vincenzo Librandi, Jul 15 2011

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 *)

PARI

a(n)=fibonacci(2*n+1)<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 */

SageMath

[2^n*fibonacci(2*n+1) for n in range(41)] # G. C. Greubel, Jul 28 2024

Formula

a(n) = Sum_{k=0..2*n} A027907(n, k)*A000045(k+1).

From Paul Barry, Jul 16 2003: (Start)

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)). (End)

From R. J. Mathar, Nov 04 2008: (Start)

G.f.: (1-2*x)/(1-6*x+4*x^2).

a(n) = 6*a(n-1) - 4*a(n-2). (End)

a(n) = Sum_{k=0..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

From Michael Somos, May 26 2014: (Start)

a(n) - a(n-1) = A069429(n).

a(n+1) * a(n-1) - a(n)^2 = 4^n.

G.f.: 1 / (1 - 4*x / (1 - x / (1 - x))). (End)

E.g.f.: exp(3*x)*(5*cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, May 24 2024