cp's OEIS Frontend

A096535 a(0) = a(1) = 1; a(n) = (a(n-1) + a(n-2)) mod n.

%I A096535 #25 Aug 28 2016 18:18:35
%S A096535 1,1,0,1,1,2,3,5,0,5,5,10,3,0,3,3,6,9,15,5,0,5,5,10,15,0,15,15,2,17,
%T A096535 19,5,24,29,19,13,32,8,2,10,12,22,34,13,3,16,19,35,6,41,47,37,32,16,
%U A096535 48,9,1,10,11,21,32,53,23,13,36,49,19,1,20,21,41,62,31,20,51,71,46,40,8,48,56
%N A096535 a(0) = a(1) = 1; a(n) = (a(n-1) + a(n-2)) mod n.
%C A096535 Suggested by _Leroy Quet_.
%C A096535 Three conjectures: (1) All numbers appear infinitely often, i.e., for every number k >= 0 and every frequency f > 0 there is an index i such that a(i) = k is the f-th occurrence of k in the sequence.
%C A096535 (2) a(j) = a(j-1) + a(j-2) and a(j) = a(j-1) + a(j-2) - j occur approximately equally often, i.e., lim_{n->infinity} x_n / y_n = 1, where x_n is the number of j <= n such that a(j) = a(j-1) + a(j-2) and y_n is the number of j <= n such that a(j) = a(j-1) + a(j-2) - j (cf. A122276).
%C A096535 (3) There are sections a(g+1), ..., a(g+k) of arbitrary length k such that a(g+h) = a(g+h-1) + a(g+h-2) for h = 1,...,k, i.e., the sequence is nondecreasing in these sections (cf. A122277, A122278, A122279). - _Klaus Brockhaus_, Aug 29 2006
%C A096535 a(A197877(n)) = n and a(m) <> n for m < A197877(n); see first conjecture. - _Reinhard Zumkeller_, Oct 19 2011
%H A096535 T. D. Noe, <a href="/A096535/b096535.txt">Table of n, a(n) for n = 0..10000</a>
%t A096535 l = {1, 1}; For[i = 2, i <= 100, i++, len = Length[l]; l = Append[l, Mod[l[[len]] + l[[len - 1]], i]]]; l
%t A096535 f[s_] := f[s] = Append[s, Mod[s[[ -2]] + s[[ -1]], Length[s]]]; Nest[f, {1, 1}, 80] (* _Robert G. Wilson v_, Aug 29 2006 *)
%t A096535 RecurrenceTable[{a[0]==a[1]==1,a[n]==Mod[a[n-1]+a[n-2],n]},a,{n,90}] (* _Harvey P. Dale_, Apr 12 2013 *)
%o A096535 (Haskell)
%o A096535 a096535 n = a096535_list !! n
%o A096535 a096535_list = 1 : 1 : f 2 1 1 where
%o A096535    f n x x' = y : f (n+1) y x where y = mod (x + x') n
%o A096535 -- _Reinhard Zumkeller_, Oct 19 2011
%Y A096535 Cf. A079777, A096274 (location of 0's), A096534, A132678.
%K A096535 easy,nonn,nice
%O A096535 0,6
%A A096535 _Franklin T. Adams-Watters_, Jun 23 2004