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A082761
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Trinomial transform of the Fibonacci numbers (A000045).
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7
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Hankel transform of sum{k=0..n, (-1)^k*C(2k,k)} (see A054108). [From Paul Barry (pbarry(AT)wit.ie), Jan 13 2009]
Hankel transform of A046748. [From Paul Barry (pbarry(AT)wit.ie), 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. [From John M. Campbell, Jul 12, 2011]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..150
Index to sequences with linear recurrences with constant coefficients, signature (6,-4).
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FORMULA
| a(n) = Sum[ Trinomial[n, k] Fibonacci[k+1], {k, 0, 2n} ] where Trinomial[n, k] = trinomial coefficients (A027907)
a(n) = 2^n* Fibonacci(2*n+1)
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). [From R. J. Mathar, Nov 04 2008]
a(n)=Sum_{k, 0<=k<=n}A147703(n,k)*3^k. [From Philippe DELEHAM, Nov 14 2008]
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PROG
| (PARI) a(n)=fibonacci(2*n+1)<<n \\ Charles R Greathouse IV, Jul 15 2011
(MAGMA) [2^n *Fibonacci(2*n+1): n in [0..40]]; // Vincenzo Librandi, Jul 15 2011
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CROSSREFS
| Sequence in context: A168606 A155485 A155181 * A076035 A120978 A104550
Adjacent sequences: A082758 A082759 A082760 * A082762 A082763 A082764
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KEYWORD
| easy,nonn
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AUTHOR
| Emanuele Munarini (munarini(AT)mate.polimi.it), May 21 2003
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