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Search: id:A128429
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| A128429 |
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A linear recurrence sequence: a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-6). |
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2
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| 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
(list; graph; listen)
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OFFSET
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0,7
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COMMENT
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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.
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REFERENCES
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Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002
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LINKS
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Eric Weisstein's World of Mathematics, Golden Ratio
Bruce Rawles, Sacred Geometry
Kelley L. Ross, The Golden Ratio and The Fibonacci Numbers
Wikipedia Golden Ratio
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FORMULA
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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 (mathar(AT)strw.leidenuniv.nl), Nov 16 2007
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CROSSREFS
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Cf. Fibonacci numbers A000045; Lucas numbers A000032.
Sequence in context: A092863 A138694 A115288 this_sequence A131500 A003461 A023375
Adjacent sequences: A128426 A128427 A128428 this_sequence A128430 A128431 A128432
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KEYWORD
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nonn
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AUTHOR
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Luis A Restrepo (luisiii(AT)mac.com), Mar 05 2007
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EXTENSIONS
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Edited by Dean Hickerson (dean(AT)math.ucdavis.edu) and Don Reble (djr(AT)nk.ca), Mar 09 2007
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