A128429 A linear recurrence sequence: a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-6).

1, 1, 1, 1, 1, 1, 4, 7, 10, 16, 25, 40, 67, 109, 175, 283, 457, 739, 1198, 1939, 3136, 5074, 8209, 13282, 21493, 34777, 56269, 91045, 147313, 238357, 385672, 624031, 1009702, 1633732, 2643433, 4277164, 6920599, 11197765, 18118363, 29316127, 47434489

Offset: 0

Author

Luis A Restrepo (luisiii(AT)mac.com), Mar 05 2007

Keywords

• nonn

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).

Crossrefs

Cf. Fibonacci numbers A000045; Lucas numbers A000032.

Programs

Mathematica

LinearRecurrence[{1, 0, 1, 0, 1, 1}, {1, 1, 1, 1, 1, 1}, 41] (* Jean-François Alcover, Sep 25 2017 *)

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

Extensions

Edited by Dean Hickerson and Don Reble, Mar 09 2007