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%I #25 Dec 27 2019 18:23:50
%S 0,1,4,81,3136,10609
%N Square tribonacci numbers.
%C These are the only square tribonacci numbers having indices < 47000.
%C Next term, if it exists, is too large to present here. - _Robert G. Wilson v_, Apr 24 2007
%C Indices of the square tribonacci numbers: 1,4,9,15,17.
%C The square Fibonacci numbers seem to be even rarer, namely just 1 & 144. - _Robert G. Wilson v_, Apr 24 2007
%C It is very likely that there are no further terms. - _N. J. A. Sloane_, Apr 25 2007
%C 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
%H Attila Pethö, <a href="http://www.emis.de/journals/AUSM/C2-1/math21-5.pdf">Fifteen problems in number theory</a>, Acta Universitatis Sapientiae. Mathematica (2010) Volume: 2, Issue: 1, page 72-83. See Problem 1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TribonacciNumber.html">Tribonacci Number</a>
%e 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.
%t 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 *)
%t Drop[Select[LinearRecurrence[{1,1,1},{0,1,1},20],IntegerQ[Sqrt[#]]&],2] (* _Harvey P. Dale_, Mar 17 2017 *)
%Y Intersection of A000073 and A000290.
%K nonn
%O 1,3
%A _David A. G. Gillies_, Apr 23 2007
%E Edited by _Robert G. Wilson v_, Apr 24 2007
%E More terms from _T. D. Noe_, Jun 22 2007
%E a(1) = 0 inserted by _Felix Fröhlich_, Dec 11 2019