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A128911 Square tribonacci numbers.

%I A128911 #26 Feb 16 2025 08:33:05
%S A128911 0,1,4,81,3136,10609
%N A128911 Square tribonacci numbers.
%C A128911 These are the only square tribonacci numbers having indices < 47000.
%C A128911 Next term, if it exists, is too large to present here. - _Robert G. Wilson v_, Apr 24 2007
%C A128911 Indices of the square tribonacci numbers: 1,4,9,15,17.
%C A128911 The square Fibonacci numbers seem to be even rarer, namely just 1 & 144. - _Robert G. Wilson v_, Apr 24 2007
%C A128911 It is very likely that there are no further terms. - _N. J. A. Sloane_, Apr 25 2007
%C A128911 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 A128911 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 A128911 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TribonacciNumber.html">Tribonacci Number</a>
%e A128911 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 A128911 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 A128911 Drop[Select[LinearRecurrence[{1,1,1},{0,1,1},20],IntegerQ[Sqrt[#]]&],2] (* _Harvey P. Dale_, Mar 17 2017 *)
%Y A128911 Intersection of A000073 and A000290.
%K A128911 nonn
%O A128911 1,3
%A A128911 _David A. G. Gillies_, Apr 23 2007
%E A128911 Edited by _Robert G. Wilson v_, Apr 24 2007
%E A128911 More terms from _T. D. Noe_, Jun 22 2007
%E A128911 a(1) = 0 inserted by _Felix Fröhlich_, Dec 11 2019