A223487 Number of missing residues in Lucas sequence mod n.

0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 4, 2, 1, 0, 8, 5, 1, 7, 7, 10, 8, 8, 4, 10, 13, 2, 0, 8, 19, 16, 12, 10, 16, 14, 22, 21, 9, 25, 15, 30, 22, 16, 10, 24, 28, 25, 32, 31, 12, 26, 20, 16, 9, 25, 39, 28, 28, 38, 22, 42, 33, 41, 30, 22, 49, 32, 16, 42, 36, 44, 27, 55

Offset: 1

Author

Casey Mongoven, Mar 20 2013

Keywords

• nonn

Comments

The Lucas numbers mod n for any n are periodic - see A106291 for period lengths.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.

Crossrefs

Cf. A118965.

Programs

Mathematica

pisano[n_] := Module[{a = {2, 1}, a0, k = 0, s, t}, If[n == 1, 1, a0 = a; t = a; While[k++; s = Mod[Plus @@ a, n]; AppendTo[t, s]; a[[1]] = a[[2]]; a[[2]] = s; a != a0]; t]]; Join[{0, 0}, Table[u = Union[pisano[n]]; mx = Max[u]; Length[Complement[Range[0, mx], u]], {n, 3, 100}]] (* T. D. Noe, Mar 22 2013 *)