A132678 -id:A132678 - 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 *)

A096274 Indices of zeros in A096535.

Original entry on oeis.org

2, 8, 13, 20, 25, 595, 1044, 7932, 74247, 14693476, 16766626, 24072338, 72643740, 1881945888, 3304284638, 5163731431, 5669949197, 16209038688, 23714508403, 56796564073, 181057353263, 323874989643, 406930606305, 539293061152, 1751203649485, 2136659012156

Offset: 1

Jim Nastos, Jun 24 2004

nonn

Suggested by Leroy Quet.

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

Cf. A132678.

/* C program from Peter Pein */
#include 
main(int argc, char *argv[])
{ long long a0=1, a1=1, n=1, tmp, nmax;
if (argc != 2) { fprintf(stderr,"%s n\ncalculates the indices of the first n zeros in A096535\n", argv[0]);
return(1); }
nmax=atol(argv[1]);
while (nmax-- > 0) {
while(a1 != 0) {
tmp = (a0 + a1) % ++n; a0 = a1; a1 = tmp; }
printf("%lld\n",n++); a1 = a0; a0 = 0; }
return 0; }

import Data.List (elemIndices)
a096274 n = a096274_list !! (n-1)
a096274_list = elemIndices 0 a096535_list
-- Reinhard Zumkeller, Oct 19 2011

a = b = 1; lst = {}; Do[c = Mod[a + b, n]; If[c == 0, AppendTo[lst, n]; Print@n]; a = b; b = c, {n, 2, 10^9}] (* Robert G. Wilson v, Dec 17 2007 *)

a(13) from Robert G. Wilson v, Jun 23 2004
a(14) - a(16) from Robert G. Wilson v, Aug 30 2006
Extended to a(26) by Zak Seidov, Peter Pein (petsie(AT)dordos.net) and Martin Fuller, Nov 22 2007

A197877 Smallest number m such that A096535(m) = n.

Original entry on oeis.org

2, 0, 5, 6, 465, 7, 16, 208, 37, 17, 11, 58, 40, 35, 84, 18, 45, 29, 395, 30, 68, 59, 41, 62, 32, 191, 325, 109, 369, 33, 89, 72, 36, 85, 42, 47, 64, 51, 101, 88, 77, 49, 125, 394, 1124, 249, 76, 50, 54, 65, 119, 74, 2193, 61, 483, 133, 80, 186, 95, 990, 468

Offset: 0

Reinhard Zumkeller, Oct 19 2011

nonn

A096535(a(n)) = n and A096535(m) <> n for m < a(n), concerning first conjecture in A096535.

			a(0) = A096274(1) = 2; a(1) = A132678(1) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a197877 n = a197877_list !! n
a197877_list = map (fromJust . (`elemIndex` a096535_list)) [0..]

cp's OEIS Frontend

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

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A096274 Indices of zeros in A096535.

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A197877 Smallest number m such that A096535(m) = n.

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