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A082761
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Trinomial transform of the Fibonacci numbers (A000045).
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10
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1, 4, 20, 104, 544, 2848, 14912, 78080, 408832, 2140672, 11208704, 58689536, 307302400, 1609056256, 8425127936, 44114542592, 230986743808, 1209462292480, 6332826779648, 33159111507968, 173623361929216, 909103725543424
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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).
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FORMULA
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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
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EXAMPLE
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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 + ...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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easy,nonn
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AUTHOR
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Emanuele Munarini, May 21 2003
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STATUS
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approved
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