A096534 -id:A096534 - cp's OEIS frontend

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

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

Franklin T. Adams-Watters, Jun 23 2004

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

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A079777, A096274 (location of 0's), A096534, A132678.

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

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 *)

A079777 a(0) = 0, a(1) = 1; for n > 1, a(n) = (a(n-1) + a(n-2)) (mod n).

Original entry on oeis.org

0, 1, 1, 2, 3, 0, 3, 3, 6, 0, 6, 6, 0, 6, 6, 12, 2, 14, 16, 11, 7, 18, 3, 21, 0, 21, 21, 15, 8, 23, 1, 24, 25, 16, 7, 23, 30, 16, 8, 24, 32, 15, 5, 20, 25, 0, 25, 25, 2, 27, 29, 5, 34, 39, 19, 3, 22, 25, 47, 13, 0, 13, 13, 26, 39, 0, 39, 39, 10, 49, 59, 37, 24, 61, 11, 72, 7, 2, 9, 11, 20

Offset: 0

Joseph L. Pe, Mar 08 2003

nonn

Robert G. Wilson v, Table of n, a(n) for n = 0..10001 [a(9917) corrected by _Georg Fischer_, Dec 03 2023]

Cf. A000045, A058981, A096534, A096535, Zeros in A073853.

l = {1, 1}; For[i = 3, 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, {0, 1}, 80] (* Robert G. Wilson v, Dec 16 2007 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==Mod[a[n-1]+a[n-2],n]},a,{n,80}] (* Harvey P. Dale, Nov 29 2019 *)

Edited by Robert G. Wilson v, Dec 16 2007

A072987 FIBMOD numbers: a(1) = a(2) = 1, a(n) = a(n-1) mod (n-1) + a(n-2) mod (n-2).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 6, 7, 13, 10, 13, 11, 12, 23, 20, 12, 16, 28, 25, 14, 19, 33, 29, 15, 20, 35, 28, 8, 8, 16, 24, 40, 31, 38, 34, 37, 34, 34, 68, 62, 49, 28, 35, 63, 53, 25, 32, 57, 40, 48, 88, 84, 67, 44, 57, 45, 46, 91, 78, 50, 68, 56, 62, 118, 115, 102

Offset: 1

Benoit Cloitre, Aug 14 2002

Superseeker suggested that this sequence might be related to A096534 via the transformation T026 (coefficients of Sn(z)/(1+z)) for a(3) to a(76). - Eli Jaffe, Sep 16 2015

Superseeker's reply seems to be true: it appears that the present sequence has generating function equal to (1+x)*(1+x*G(X)), where G(x) is the g.f. for A096534. - N. J. A. Sloane, Nov 23 2015

			For n=8, a(8) = (a(7) mod 7) + (a(6) mod 6) = 1 + 5 = 6. - _Eli Jaffe_, Sep 16 2015

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Cf. A096534.

I:=[1, 1]; [n le 2 select I[n] else Self(n-1) mod (n-1) + Self(n-2) mod (n-2): n in [1..80]]; // Vincenzo Librandi, Sep 26 2015

a:= proc(n) option remember; `if`(n<3, 1,
      add(irem(a(n-j), n-j), j=1..2))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 20 2018

a = {1, 1}; Do[AppendTo[a, Mod[a[[n - 1]], n - 1] + Mod[a[[n - 2]], n - 2]], {n, 3, 76}]; a (* Michael De Vlieger, Sep 17 2015 *)
RecurrenceTable[{a[1]==a[2]==1,a[n]==Mod[a[n-1],n-1]+Mod[a[n-2],n-2]},a,{n,80}] (* Harvey P. Dale, Oct 28 2017 *)

a=vector(10^5); a[1]=a[2]=1; for(n=3, #a, a[n]=a[n-1]%(n-1)+a[n-2]%(n-2)); a \\ Altug Alkan, Mar 20 2018

a(n) < 2n.

A096534(n) == a(n) mod n. - Danny Rorabaugh, Oct 13 2015

Corrected by Harvey P. Dale, Oct 28 2017

cp's OEIS Frontend

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

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A079777 a(0) = 0, a(1) = 1; for n > 1, a(n) = (a(n-1) + a(n-2)) (mod n).

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A072987 FIBMOD numbers: a(1) = a(2) = 1, a(n) = a(n-1) mod (n-1) + a(n-2) mod (n-2).

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