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

0,7

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

Cf. Fibonacci numbers A000045; Lucas numbers A000032.

Sequence in context: A153003 A213484 A362255 * A191154 A209257 A131500

Adjacent sequences: A128426 A128427 A128428 * A128430 A128431 A128432

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