A284786 Pisano period of sequence A006054 modulo n.

1, 7, 26, 14, 62, 182, 42, 28, 78, 434, 266, 182, 12, 42, 806, 56, 614, 546, 254, 434, 546, 266, 1106, 364, 310, 84, 234, 42, 28, 5642, 1986, 112, 3458, 4298, 1302, 546, 2814, 1778, 156, 868, 40, 546, 42, 266, 2418, 1106, 4514, 728, 294, 2170, 7982, 84, 5726, 1638, 8246, 84, 3302, 28, 7082, 5642, 582

Offset: 1

Author

Patrick D McLean, Apr 02 2017

Keywords

• nonn

Links

• Robert Israel, Mathematics StackExchange, When does x^3-x^2-2x+1 split mod p.
• Wikipedia, Pisano period.

Crossrefs

Cf. A001175 Pisano periods of Fibonacci numbers mod n.

Cf. A045472.

Programs

Maple

f:= proc(n) option remember; local F, t, k, a;
F:= ifactors(n)[2];
if nops(F) > 1 then
  return(ilcm(seq(procname(t[1]^t[2]),t=F)))
fi;
a:= [0,0,1];
for k from 1 do
  a:= [a[2],a[3],2*a[3]+a[2]-a[1] mod n];
  if  a = [0,0,1] then return k fi;
od:
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Jun 14 2017

Mathematica

Table[FindTransientRepeat[
    Mod[LinearRecurrence[{2, 1, -1}, {0, 0, 1}, 100000], n], 2] //
   Last // Length, {n, 1, 20}]

Formula

From Robert Israel, Jun 14 2017: (Start)

If m and n are coprime, a(m*n) = lcm(a(m),a(n)).

If p is a prime such that the polynomial x^3-x^2-2*x+1 splits into distinct factors mod p, then a(p) divides p-1. These primes are A045472. (End)

Extensions

More terms from Robert Israel, Jun 14 2017