The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A082761 Trinomial transform of the Fibonacci numbers (A000045). 10
 1, 4, 20, 104, 544, 2848, 14912, 78080, 408832, 2140672, 11208704, 58689536, 307302400, 1609056256, 8425127936, 44114542592, 230986743808, 1209462292480, 6332826779648, 33159111507968, 173623361929216, 909103725543424 (list; graph; refs; listen; history; text; internal format)
 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)<

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 22 18:49 EDT 2023. Contains 361433 sequences. (Running on oeis4.)