OFFSET
0,7
COMMENTS
The characteristic polynomial of this recurrence is x^6 - x^5 - x^3 - x - 1 = (x^2 - x - 1)*(x^6 - 1)/(x^2 - 1), so the sequence can be written as the sum of a Fibonacci sequence and a sequence of period 6; see the formula line. Hence the ratio a(n+1)/a(n) has the same limit as the Fibonacci sequence does, namely the golden ratio, (1+sqrt(5))/2, about 1.61803398874989484820.
REFERENCES
Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002.
LINKS
Jinyuan Wang, Table of n, a(n) for n = 0..1000
Bruce Rawles, Sacred Geometry
Kelley L. Ross, The Golden Ratio and The Fibonacci Numbers
Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Golden Ratio
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 0, 1, 1).
FORMULA
a(n) = (1/4)*(3F(n-1) + b(n mod 6)), where F(n) = A000045(n) is the n-th Fibonacci number and b(0)=b(2)=b(3)=1, b(1)=4, b(4)=-2 and b(5)=-5.
G.f.: (-1 + x^3 + x^4 + 2*x^5)/((x^2 + x - 1)*(1 + x + x^2)*(x^2 - x + 1)). - R. J. Mathar, Nov 16 2007
MATHEMATICA
LinearRecurrence[{1, 0, 1, 0, 1, 1}, {1, 1, 1, 1, 1, 1}, 41] (* Jean-François Alcover, Sep 25 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Luis A Restrepo (luisiii(AT)mac.com), Mar 05 2007
EXTENSIONS
Edited by Dean Hickerson and Don Reble, Mar 09 2007
STATUS
approved