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A096535
a(0) = a(1) = 1; a(n) = (a(n-1) + a(n-2)) mod n.
17
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, 19, 5, 24, 29, 19, 13, 32, 8, 2, 10, 12, 22, 34, 13, 3, 16, 19, 35, 6, 41, 47, 37, 32, 16, 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
OFFSET
0,6
COMMENTS
Suggested by Leroy Quet.
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.
(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).
(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
a(A197877(n)) = n and a(m) <> n for m < A197877(n); see first conjecture. - Reinhard Zumkeller, Oct 19 2011
MATHEMATICA
l = {1, 1}; For[i = 2, i <= 100, i++, len = Length[l]; l = Append[l, Mod[l[[len]] + l[[len - 1]], i]]]; l
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 *)
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 *)
PROG
(Haskell)
a096535 n = a096535_list !! n
a096535_list = 1 : 1 : f 2 1 1 where
f n x x' = y : f (n+1) y x where y = mod (x + x') n
-- Reinhard Zumkeller, Oct 19 2011
CROSSREFS
Cf. A079777, A096274 (location of 0's), A096534, A132678.
Sequence in context: A254271 A082118 A079344 * A126047 A023049 A240979
KEYWORD
easy,nonn,nice
AUTHOR
STATUS
approved