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 A033889 a(n) = Fibonacci(4*n + 1). 14
 1, 5, 34, 233, 1597, 10946, 75025, 514229, 3524578, 24157817, 165580141, 1134903170, 7778742049, 53316291173, 365435296162, 2504730781961, 17167680177565, 117669030460994, 806515533049393, 5527939700884757, 37889062373143906, 259695496911122585, 1779979416004714189 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For positive n, a(n) equals (-1)^n times the permanent of the (4n) X (4n) tridiagonal matrix with sqrt(i)'s along the three central diagonals, where i is the imaginary unit. - John M. Campbell, Jul 12 2011 a(n) = 5^n*a(n;3/5) = (16/5)^n*a(2n;3/4), and F(4n)=5^n*b(n;3/5) = (16/5)^n*b(2n;3/4), where a(n;d) and b(n;d), n=0,1,...,d \in C, denote the delta-Fibonacci numbers defined in comments to A014445. Two of these identities from the following relations follows: F(k+1)^n*a(n;F(k)/F(k+1))=F(kn+1), and F(k+1)^n*b(n;F(k)/F(k+1))=F(kn) (see also Witula's et al. papers). - Roman Witula, Jul 24 2012 REFERENCES D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers, (submitted to Fibonacci Quart. 2012). LINKS Bruno Berselli, Table of n, a(n) for n = 0..500 Tanya Khovanova, Recursive Sequences R. Witula, Binomials Transformation Formulae of Scaled Lucas Numbers, Demonstratio Math, , Vol. XLVI No 1 2013. R. Witula, Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042 Index entries for linear recurrences with constant coefficients, signature (7,-1) FORMULA a(n) = 7*a(n-1) - a(n-2). - Floor van Lamoen, Dec 10 2001 From R. J. Mathar, Jan 17 2008: (Start) O.g.f.: (1 - 2*x)/(1 - 7*x + x^2). a(n) = A004187(n+1) - 2*A004187(n-1). (End) a(n) = (1/2)*(7/2-(3/2)*sqrt(5))^n - (1/10)*(7/2-(3/2)*sqrt(5))^n*sqrt(5)+(1/10)*sqrt(5)*(7/2+(3/2)*sqrt(5))^n + (1/2)*(7/2+(3/2)*sqrt(5))^n. - Paolo P. Lava, Jun 25 2008 a(n) = A167816(4*n+1). - Reinhard Zumkeller, Nov 13 2009 a(n) = sqrt(1+2*Fibonacci(2*n)*Fibonacci(2*n+1)+5*(Fibonacci(2*n)*Fibonacci(2*n+1))^2). - Artur Jasinski, Feb 06 2010 a(n) = Sum_{k=0..n} A122070(n,k)*2^k. - Philippe Deléham, Mar 13 2012 a(n) = Fibonacci(2*n)^2 + Fibonacci(2*n)*Fibonacci(2*n+2) + 1. - Gary Detlefs, Apr 18 2012 a(n) = Fibonacci(2n)^2 + Fibonacci(2n+1)^2. - Bruno Berselli, Apr 19 2012 a(n) = Sum_{k = 0..n} A238731(n,k)*4^k. - Philippe Deléham, Mar 05 2014 a(n) = A000045(A016813(n)). - Michel Marcus, Mar 05 2014 2*a(n) = Fibonacci(4*n) + Lucas(4*n). - Bruno Berselli, Oct 13 2017 MATHEMATICA Table[Fibonacci[4*n+1], {n, 0, 14}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2008 *) PROG (MAGMA) [Fibonacci(4*n+1): n in [0..100]]; // Vincenzo Librandi, Apr 16 2011 (PARI) a(n)=fibonacci(4*n+1) \\ Charles R Greathouse IV, Jul 15 2011 (PARI) Vec((1-2*x)/(1-7*x+x^2) + O(x^100)) \\ Altug Alkan, Dec 10 2015 CROSSREFS Cf. A000032, A000045, A004187, A014445, A016813, A122070, A167816, Cf. A081018, A081016, A172968. Cf. A049684, A081068. Sequence in context: A015545 A102436 A291027 * A120469 A180909 A183415 Adjacent sequences:  A033886 A033887 A033888 * A033890 A033891 A033892 KEYWORD nonn,easy AUTHOR STATUS approved

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