A106286 Number of orbits of the 4-step recursion mod n.

1, 4, 6, 28, 3, 24, 10, 220, 91, 12, 130, 240, 343, 40, 168, 1756, 19, 364, 22, 132, 81, 2068, 26, 1968, 253, 1372, 2336, 448, 2557, 672, 16, 14044, 1143, 76, 108, 4612, 1411, 88, 3084, 1860, 11815, 324, 22, 32092, 13213, 104, 50, 15792, 2467, 4012, 168, 17812

Offset: 1

Author

T. D. Noe, May 02 2005

Keywords

• nonn

Comments

Consider the 4-step recursion x(k)=x(k-1)+x(k-2)+x(k-3)+x(k-4) mod n. For any of the n^4 initial conditions x(1), x(2), x(3) and x(4) in Zn, the recursion has a finite period. Each of these n^4 vectors belongs to exactly one orbit. In general, there are only a few different orbit lengths (A106289) for each n. For instance, the 220 orbits mod 8 have lengths of 1, 5, 10 and 20.

Links

• D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
• Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Crossrefs

Cf. A015134 (orbits of Fibonacci sequences), A106285 (orbits of 3-step sequences), A106287 (orbits of 5-step sequences), A106289 (number of different orbit lengths), A106308 (n producing a simple orbit structure).