A128911 Square tribonacci numbers.

0, 1, 4, 81, 3136, 10609

Offset: 1

Author

David A. G. Gillies, Apr 23 2007

Keywords

• nonn

Comments

These are the only square tribonacci numbers having indices < 47000.

Next term, if it exists, is too large to present here. - Robert G. Wilson v, Apr 24 2007

Indices of the square tribonacci numbers: 1,4,9,15,17.

The square Fibonacci numbers seem to be even rarer, namely just 1 & 144. - Robert G. Wilson v, Apr 24 2007

It is very likely that there are no further terms. - N. J. A. Sloane, Apr 25 2007

Using modular arithmetic and quadratic residues, it can be shown that there are no additional squares in the first 10^9 tribonacci numbers. - T. D. Noe, Jun 22 2007

Examples

			The terms 0, 1, 4, 81, 3136, 10609 are members of the sequence since their square roots are 0, 1, 2, 9, 56, 103 respectively.
		

Links

• Attila Pethö, Fifteen problems in number theory, Acta Universitatis Sapientiae. Mathematica (2010) Volume: 2, Issue: 1, page 72-83. See Problem 1.
• Eric Weisstein's World of Mathematics, Tribonacci Number

Crossrefs

Intersection of A000073 and A000290.

Programs

Mathematica

a = b = 0; c = 1; lst = {}; Do[{a, b, c} = {b, c, a + b + c}; If[ IntegerQ@ Sqrt@c, AppendTo[lst, c]], {n, 2, 47000}]; lst (* Robert G. Wilson v, Apr 24 2007 *)
Drop[Select[LinearRecurrence[{1,1,1},{0,1,1},20],IntegerQ[Sqrt[#]]&],2] (* Harvey P. Dale, Mar 17 2017 *)

Extensions

Edited by Robert G. Wilson v, Apr 24 2007

More terms from T. D. Noe, Jun 22 2007

a(1) = 0 inserted by Felix Fröhlich, Dec 11 2019