A118965 Number of missing residues in Fibonacci sequence mod n.

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 1, 4, 0, 0, 5, 4, 7, 7, 0, 12, 8, 4, 11, 0, 8, 0, 7, 19, 0, 12, 11, 14, 21, 0, 21, 8, 25, 14, 10, 22, 24, 10, 24, 0, 25, 32, 33, 12, 0, 16, 22, 16, 25, 43, 31, 24, 38, 22, 5, 36, 41, 40, 22, 20, 28, 16, 48, 40, 0, 27, 57

Offset

1,8

Comments

If n belongs to A079002, then a(n) = 0. - Michel Marcus, May 27 2013

a(n) = number of zeros in n-th row of triangle A128924. - Reinhard Zumkeller, Jan 16 2014

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cyrus Hsia et al., Mathematical Mayhem Editors, Fibonacci residues, Crux Mathematicorum, 1997 Vol. 23 No. 4, pp. 224-226.

Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.

D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly (67 #6, Jun-Jul 1960), pp. 525-532.

Formula

a(n) = n - A066853(n). - Michel Marcus, May 27 2013

Example

The Fibonacci sequence mod 8 is { 0 1 1 2 3 5 0 5 5 2 7 1 0 1 1 ... } - a periodic sequence with a period of 12 (see A001175). Two residues do not occur in this sequence (4 and 6), therefore a(8) = 2.

Mathematica

With[{fibs=Fibonacci[Range[300]]}, Table[Length[Complement[Range[0, n-1], Union[ Mod[fibs, n]]]], {n, 80}]] (* Harvey P. Dale, Jul 01 2021 *)

Prog

(Haskell)
a118965 = sum . map (0 ^) . a128924_row
-- Reinhard Zumkeller, Jan 16 2014
(PARI) a(n)=if(n<8, return(0)); my(v=List([1, 2])); while(v[#v] || v[#v-1]!=1, listput(v, (v[#v]+v[#v-1])%n)); n-#Set(v) \\ Charles R Greathouse IV, Jun 20 2017

Crossrefs

Cf. A066853, A001175.

Sequence in context: A238858 A106235 A288098 * A252729 A121552 A350785

Adjacent sequences: A118962 A118963 A118964 * A118966 A118967 A118968

Keyword

nonn

Author

Casey Mongoven, May 07 2006

Extensions

Offset corrected by Michel Marcus, May 27 2013

Status

approved