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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

MATHEMATICA

Table[Fibonacci[4*n+1], {n, 0, 14}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2008 *)

Table[Sqrt[1 + 2 m + 5 m^2] /. m -> Fibonacci[2 n] Fibonacci[2 n + 1], {n, 0, 20}] (* Artur Jasinski, Feb 06 2010 *)

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. A081018, A081016, A172968.

Cf. A049684, A081068.

Sequence in context: A024063 A015545 A102436 * A120469 A180909 A183415

Adjacent sequences:  A033886 A033887 A033888 * A033890 A033891 A033892

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 24 23:06 EDT 2017. Contains 284035 sequences.