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 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
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
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
EXAMPLE
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 + ...
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
(SageMath) [2^n*fibonacci(2*n+1) for n in range(41)] # G. C. Greubel, Jul 28 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Emanuele Munarini, May 21 2003
STATUS
approved