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A049666 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 zeroes of odd lengths.  - Milan Janjic, Jan 28 2015

LINKS

Table of n, a(n) for n=0..18.

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) = 11a(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: A110398 A176595 A067218 * A163462 A041222 A097708

Adjacent sequences:  A049663 A049664 A049665 * A049667 A049668 A049669

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified March 24 22:17 EDT 2017. Contains 284035 sequences.