login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A033889 a(n) = Fibonacci(4*n + 1). 15
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(2*n; 3/4), and F(4*n) = 5^n*b(n; 3/5) = (16/5)^n*b(2*n; 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(k*n+1) and F(k+1)^n*b(n; F(k)/F(k+1)) = F(k*n) (see also Witula's et al. papers). - Roman Witula, Jul 24 2012

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..500

Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Mathematics, 15(1) (2017), 477-485.

Tanya Khovanova, Recursive Sequences

Roman Witula, Binomials Transformation Formulae of Scaled Lucas Numbers, Demonstratio Math, , Vol. XLVI No 1 2013.

Roman Witula and 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) for 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). (End); corrected by Klaus Purath, Jul 29 2020

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(2*n)^2 + Fibonacci(2*n+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

a(n) = A094567(n-1) + A094567(n), assuming A094567(-1) = 0. - Klaus Purath, Jul 29 2020

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, A094567.

Cf. A049684, A081018, A081016, A081068, A172968.

Sequence in context: A015545 A102436 A291027 * A334608 A120469 A180909

Adjacent sequences:  A033886 A033887 A033888 * A033890 A033891 A033892

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 20 23:28 EDT 2020. Contains 337910 sequences. (Running on oeis4.)