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A106285
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Number of orbits of the 3-step recursion mod n.
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4
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1, 4, 3, 12, 5, 12, 9, 44, 21, 20, 25, 36, 15, 66, 15, 172, 53, 84, 21, 60, 27, 144, 23, 132, 105, 116, 183, 482, 177, 60, 91, 684, 75, 420, 45, 252, 109, 162, 45, 220, 125, 198, 265, 520, 105, 92, 2259, 516, 359, 420, 159, 884, 2867, 732, 125, 3714, 63, 1408, 59, 180
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Consider the 3-step recursion x(k)=x(k-1)+x(k-2)+x(k-3) mod n. For any of the n^3 initial conditions x(1), x(2) and x(3) in Zn, the recursion has a finite period. Each of these n^3 vectors belongs to exactly one orbit. In general, there are only a few different orbit lengths (A106288) for each n. For instance, the orbits mod 8 have lengths of 1, 2, 4, 8, 16. Interestingly, for n=2^k and n=3^k, the number of orbits appear to be A039301 and A054879, respectively.
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REFERENCES
| D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, Vol. 67, 1960, 525-532.
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LINKS
| Eric Weisstein's World of Mathematics, Fibonacci n-Step
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EXAMPLE
| Orbits for n=2: {(0,0,0)}, {(1,1,1)}, {(0,1,0), (1,0,1)} and {(0,0,1), (0,1,1), (1,1,0), (1,0,0)}
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CROSSREFS
| Cf. A015134 (orbits of Fibonacci sequences), A106286 (orbits of 4-step sequences), A106287 (orbits of 5-step sequences), A106288 (number of different orbit lengths), A106307 (n producing a simple orbit structure).
Sequence in context: A099377 A121844 A091512 * A193800 A061727 A055527
Adjacent sequences: A106282 A106283 A106284 * A106286 A106287 A106288
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KEYWORD
| nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), May 02 2005
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