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A118965 Number of missing residues in Fibonacci sequence mod n.

%I A118965 #19 Jul 01 2021 18:09:26
%S A118965 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,
%T A118965 14,21,0,21,8,25,14,10,22,24,10,24,0,25,32,33,12,0,16,22,16,25,43,31,
%U A118965 24,38,22,5,36,41,40,22,20,28,16,48,40,0,27,57
%N A118965 Number of missing residues in Fibonacci sequence mod n.
%C A118965 If n belongs to A079002, then a(n) = 0. - _Michel Marcus_, May 27 2013
%C A118965 a(n) = number of zeros in n-th row of triangle A128924. - _Reinhard Zumkeller_, Jan 16 2014
%H A118965 Reinhard Zumkeller, <a href="/A118965/b118965.txt">Table of n, a(n) for n = 1..10000</a>
%H A118965 Cyrus Hsia et al., Mathematical Mayhem Editors, <a href="http://math.ca/crux/v23/n4/page224-241.pdf">Fibonacci residues</a>, Crux Mathematicorum, 1997 Vol. 23 No. 4, pp. 224-226.
%H A118965 Casey Mongoven, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from175to192.pdf">Sonification of multiple Fibonacci-related sequences</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.
%H A118965 D. D. Wall, <a href="http://www.jstor.org/stable/2309169">Fibonacci series modulo m</a>, Amer. Math. Monthly (67 #6, Jun-Jul 1960), pp. 525-532.
%F A118965 a(n) = n - A066853(n). - _Michel Marcus_, May 27 2013
%e A118965 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.
%t A118965 With[{fibs=Fibonacci[Range[300]]},Table[Length[Complement[Range[0,n-1],Union[ Mod[fibs,n]]]],{n,80}]] (* _Harvey P. Dale_, Jul 01 2021 *)
%o A118965 (Haskell)
%o A118965 a118965 = sum . map (0 ^) . a128924_row
%o A118965 -- _Reinhard Zumkeller_, Jan 16 2014
%o A118965 (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
%Y A118965 Cf. A066853, A001175.
%K A118965 nonn
%O A118965 1,8
%A A118965 _Casey Mongoven_, May 07 2006
%E A118965 Offset corrected by _Michel Marcus_, May 27 2013