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%I #36 Dec 26 2023 10:59:45
%S 38,462,538,962,1038,1462,1538,1962,2038,2462,2538,2962,3038,3462,
%T 3538,3962,4038,4462,4538,4962,5038,5462,5538,5962,6038,6462,6538,
%U 6962,7038,7462,7538,7962,8038,8462,8538,8962,9038,9462,9538,9962,10038,10462
%N Numbers m such that m^2 ends in 444.
%C No square can end in more than three 4's.
%C When a square ends in exactly three identical digits, these digits are necessarily 444. - _Bernard Schott_, Oct 31 2019
%D Albert H. Beiler, "Recreations in the Theory of Numbers", Dover Publ., 2nd Ed. 1966, Chapter XV, "On The Square", p. 139. ISBN 0-486-21096-0.
%D A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 1 pp. 55 and 95-96 (1995)
%D David Wells, "Curious and Interesting Numbers", Revised Ed. Penguin Books, p. 152. ISBN 0-14-026149-4.
%H Harvey P. Dale, <a href="/A039685/b039685.txt">Table of n, a(n) for n = 1..1000</a>
%H British Mathematical Olympiad, <a href="https://bmos.ukmt.org.uk/home/bmo1-1995.pdf">1995 - Problem 1</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1). [From _Bruno Berselli_, Oct 27 2010]
%F a(2n+1) = 500n + 38 and a(2n+2) = 500n - 38.
%F From _Bruno Berselli_, Oct 27 2010: (Start)
%F a(n) = 250*n + 87*(-1)^n - 125.
%F G.f.: 2*x*(19 + 212*x + 19*x^2)/((1+x)*(1-x)^2).
%F a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3. (End)
%t Drop[ Flatten[ Table[{500n-38, 500n+38}, {n, 0, 21}]], 1] (* _Robert G. Wilson v_, Nov 27 2004 *)
%t Sqrt[#]&/@Select[Range[15000]^2,Mod[#,1000]==444&] (* or *) LinearRecurrence[{1,1,-1},{38,462,538},50] (* _Harvey P. Dale_, Dec 26 2023 *)
%Y Cf. A328886 (squares that end in 444).
%K nonn,base
%O 1,1
%A _Felice Russo_
%E More terms from _Patrick De Geest_, Jun 15 1999
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