A177051 A subsequence of the Fibonacci sequence such that a(n) is a quadratic residue mod a(n+1).

1, 2, 34, 55, 89, 233, 377, 1597, 17711, 28657, 121393, 317811, 1346269, 3524578, 5702887, 24157817, 39088169, 63245986, 433494437, 2971215073, 53316291173, 591286729879, 956722026041, 2504730781961, 4052739537881, 17167680177565, 308061521170129, 5527939700884757, 61305790721611591, 99194853094755497, 1779979416004714189

Offset: 1

Author

Michel Lagneau, Dec 09 2010

Keywords

• nonn

Examples

			2, 34 and 55 are in the sequence because L(2/34) = L(34/55) = 1 where L(a/b) is the Legendre symbol of a and b, which is defined to be 1 if a is a quadratic residue (mod b) and -1 if a is a quadratic non-residue (mod b).
		

Crossrefs

Cf. A034794, A000045.

Programs

Maple

with(combinat, fibonacci):k:=1:pr0:=fibonacci(k):for n from k+1 to 100 do:pr:=fibonacci(n):if
  quadres(pr0,pr)=1then pr0:=pr:printf(`%d, `,pr):else fi:od:

PARI

print1(k=1); for(n=3,100,t=fibonacci(n); if(issquare(Mod(k,t)), print1(", "k=t))) \\ Charles R Greathouse IV, Jan 09 2013

Extensions

Definition corrected by Michel Lagneau