login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A049666 a(n) = F(5n)/5, where F=A000045 (the Fibonacci sequence). 27
0, 1, 11, 122, 1353, 15005, 166408, 1845493, 20466831, 226980634, 2517253805, 27916772489, 309601751184, 3433536035513, 38078498141827, 422297015595610, 4683345669693537, 51939099382224517, 576013438874163224 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

From Johannes W. Meijer, Jun 12 2010: (Start)

For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. (End)

For n >=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 11's along the main diagonal and 1's along the subdiagonal and the superdiagonal. - John M. Campbell, Jul 08 2011

For n>=1, a(n) equals the number of words of length n-1 on alphabet {0,1,...,11} avoiding runs of zeros of odd lengths.  - Milan Janjic, Jan 28 2015

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..950

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (11, 1).

FORMULA

G.f.: x/(1-11*x-x^2).

a(n) = A102312(n)/5.

a(n) = 11*a(n-1) + a(n-2) for n>1, a(0)=0, a(1)=1. With a=golden ratio and b=1-a, a(n)=(a^(5n)-b^(5n))/(5*Sqrt(5)). - Mario Catalani (mario.catalani(AT)unito.it), Jul 24 2003

a(n) = F(n, 11), the n-th Fibonacci polynomial evaluated at x=11. - T. D. Noe, Jan 19 2006

a(n) = ((11+sqrt(125))^n-(11-sqrt(125))^n)/(2^n*sqrt(125)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009

Contribution from Johannes W. Meijer, Jun 12 2010: (Start)

a(2n) = 11*A049670(n), a(2n+1) = A097843(n).

a(3n+1) = A041227(5n), a(3n+2) = A041227(5n+3), a(3n+3) = 2*A041227(5n+4).

Limit(a(n+k)/a(k), k=infinity) = (A001946(n) + A049666(n)*sqrt(125))/2.

Limit(A001946(n)/A049666(n), n=infinity) = sqrt(125).

(End)

a(n) = F(n) + (-1)^n*5*F(n)^3 + 5*F(n)^5, n >= 0. See the D. Jennings formula given in a comment on A111125, where also the reference is given. - Wolfdieter Lang, Aug 31 2012

a(-n) = -(-1)^n * a(n). - Michael Somos, May 28 2014

EXAMPLE

G.f. = x + 11*x^2 + 122*x^3 + 1353*x^4 + 15005*x^5 + 166408*x^6 + ...

MATHEMATICA

Table[Fibonacci[5*n]/5, {n, 0, 100}] (* T. D. Noe, Oct 29 2009 *)

a[ n_] := Fibonacci[n, 11]; (* Michael Somos, May 28 2014 *)

PROG

(Mupad) numlib::fibonacci(5*n)/5 $ n = 0..25; // Zerinvary Lajos, May 09 2008

(Sage) from sage.combinat.sloane_functions import recur_gen3; it = recur_gen3(0, 1, 11, 11, 1, 0); [it.next() for i in xrange(1, 22)] # Zerinvary Lajos, Jul 09 2008

(Sage) [lucas_number1(n, 11, -1) for n in xrange(0, 19)] # Zerinvary Lajos, Apr 27 2009

(Sage) [fibonacci(5*n)/5 for n in xrange(0, 19)] # Zerinvary Lajos, May 15 2009

(PARI) a(n)=fibonacci(5*n)/5 \\ Charles R Greathouse IV, Feb 03 2014

CROSSREFS

A column of array A028412.

Cf. A102312. - Zerinvary Lajos, May 15 2009

Cf. A243399.

Sequence in context: A067218 A293805 A288791 * A163462 A041222 A097708

Adjacent sequences:  A049663 A049664 A049665 * A049667 A049668 A049669

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 23 13:09 EST 2017. Contains 295127 sequences.