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

1,8

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.

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 *)

Crossrefs

Cf. A118965.

Sequence in context: A182256 A164993 A305572 * A226911 A291956 A023987

Adjacent sequences: A223484 A223485 A223486 * A223488 A223489 A223490

Keyword

nonn

Author

Casey Mongoven, Mar 20 2013

Status

approved