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A033889 a(n) = Fibonacci(4*n + 1). 15

%I

%S 1,5,34,233,1597,10946,75025,514229,3524578,24157817,165580141,

%T 1134903170,7778742049,53316291173,365435296162,2504730781961,

%U 17167680177565,117669030460994,806515533049393,5527939700884757,37889062373143906,259695496911122585,1779979416004714189

%N a(n) = Fibonacci(4*n + 1).

%C 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

%C 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

%D D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers, (submitted to Fibonacci Quart. 2012).

%H Bruno Berselli, <a href="/A033889/b033889.txt">Table of n, a(n) for n = 0..500</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H R. Witula, <a href="http://demmath.mini.pw.edu.pl/archive/dm46_1/2.pdf">Binomials Transformation Formulae of Scaled Lucas Numbers</a>, Demonstratio Math, , Vol. XLVI No 1 2013.

%H R. Witula, Damian Slota, <a href="http://dx.doi.org/10.2298/AADM0902310W">delta-Fibonacci numbers</a>, Appl. Anal. Discr. Math 3 (2009) 310-329, <a href="http://www.ams.org/mathscinet-getitem?mr=2555042">MR2555042</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-1)

%F a(n) = 7*a(n-1) - a(n-2). - _Floor van Lamoen_, Dec 10 2001

%F From _R. J. Mathar_, Jan 17 2008: (Start)

%F O.g.f.: (1 - 2*x)/(1 - 7*x + x^2).

%F a(n) = A004187(n+1) - 2*A004187(n-1). (End)

%F 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

%F a(n) = A167816(4*n+1). - _Reinhard Zumkeller_, Nov 13 2009

%F 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

%F a(n) = Sum_{k=0..n} A122070(n,k)*2^k. - _Philippe Deléham_, Mar 13 2012

%F a(n) = Fibonacci(2*n)^2 + Fibonacci(2*n)*Fibonacci(2*n+2) + 1. - _Gary Detlefs_, Apr 18 2012

%F a(n) = Fibonacci(2n)^2 + Fibonacci(2n+1)^2. - _Bruno Berselli_, Apr 19 2012

%F a(n) = Sum_{k = 0..n} A238731(n,k)*4^k. - _Philippe Deléham_, Mar 05 2014

%F a(n) = A000045(A016813(n)). - _Michel Marcus_, Mar 05 2014

%F 2*a(n) = Fibonacci(4*n) + Lucas(4*n). - _Bruno Berselli_, Oct 13 2017

%t Table[Fibonacci[4*n+1], {n,0,14}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2008 *)

%o (MAGMA) [Fibonacci(4*n+1): n in [0..100]]; // _Vincenzo Librandi_, Apr 16 2011

%o (PARI) a(n)=fibonacci(4*n+1) \\ _Charles R Greathouse IV_, Jul 15 2011

%o (PARI) Vec((1-2*x)/(1-7*x+x^2) + O(x^100)) \\ _Altug Alkan_, Dec 10 2015

%Y Cf. A000032, A000045, A004187, A014445, A016813, A122070, A167816,

%Y Cf. A081018, A081016, A172968.

%Y Cf. A049684, A081068.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

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Last modified November 11 18:50 EST 2019. Contains 329031 sequences. (Running on oeis4.)