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

Author

Casey Mongoven, May 07 2006

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

Cf. A066853, A001175.

Programs

Haskell

a118965 = sum . map (0 ^) . a128924_row
-- Reinhard Zumkeller, Jan 16 2014

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 *)

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

Formula

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

Extensions

Offset corrected by Michel Marcus, May 27 2013