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A109344 a(n) consists of n 4's, n-1 8's and a single 9 (in that order).

%I A109344 #53 Jul 20 2025 12:37:36
%S A109344 49,4489,444889,44448889,4444488889,444444888889,44444448888889,
%T A109344 4444444488888889,444444444888888889,44444444448888888889,
%U A109344 4444444444488888888889,444444444444888888888889,44444444444448888888888889,4444444444444488888888888889,444444444444444888888888888889
%N A109344 a(n) consists of n 4's, n-1 8's and a single 9 (in that order).
%C A109344 All terms are squares. The square roots are in A067275.
%C A109344 In the same category, we have 729, 71289, 7112889, ...  with square roots in A199688. - _Michel Marcus_, Mar 21 2014
%D A109344 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 30 at p. 61.
%D A109344 Italo Ghersi, Matematica dilettevole e curiosa, p. 112, Hoepli, Milano, 1967. [From _Vincenzo Librandi_, Dec 31 2008]
%D A109344 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Example 5.5 on page 159.
%D A109344 Paul Zeitz, The Art and Craft of Problem Solving, John Wiley and Sons, Inc., New York, 1999.
%H A109344 Seiichi Manyama, <a href="/A109344/b109344.txt">Table of n, a(n) for n = 1..500</a>
%H A109344 Emile Fourrey, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k875411n/f82.image">Récréations arithmétiques</a>, Vuibert, 1899 and after, Paris, pages 72-73.
%H A109344 Michael Penn, <a href="https://www.youtube.com/watch?v=5Z8OXvbJftQ">This is always a perfect square??</a>, YouTube video, 2021.
%H A109344 StackExchange, <a href="http://math.stackexchange.com/questions/16946">History of 'Show that 44...88...9 is a perfect square'</a>.
%H A109344 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (111,-1110,1000).
%F A109344 a(1)=49; a(n) = 4*(Sum_{i=n..2*n-1} 10^i) + 8*(Sum_{i=1..n-1} 10^i) + 9, n >= 2.
%F A109344 From _R. J. Mathar_, Jan 06 2009: (Start)
%F A109344 a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) = (4*100^n + 4*10^n + 1)/9.
%F A109344 G.f.: x*(49 - 950*x + 1000*x^2)/((1-x)*(100*x-1)*(10*x-1)). (End)
%F A109344 E.g.f.: (1/9)*exp(x)*(1 + 4*exp(9*x) + 4*exp(99*x)) - 1. - _Stefano Spezia_, Aug 22 2019
%e A109344 a(5) = 4444488889 because the first 5 digits are 4's, the next 5 - 1 = 4 digits are 8's and the last digit is 9.
%p A109344 a:=n->4*sum('10^i', 'i'=n..2*n-1)+8*sum('10^i', 'i'=1..n-1)+9;
%t A109344 LinearRecurrence[{111,-1110,1000},{49,4489,444889},20] (* _Harvey P. Dale_, Nov 28 2014 *)
%o A109344 (PARI) a(n) = (2*10^n/3 + 1/3)^2 \\ _David A. Corneth_, Jan 27 2021
%Y A109344 Cf. A067275, A199688.
%K A109344 nonn,base,easy
%O A109344 1,1
%A A109344 Nicholas Protonotarios (protost(AT)hotmail.com), Aug 21 2005
%E A109344 More terms from _Harvey P. Dale_, Nov 28 2014
%E A109344 Edited by _Jon E. Schoenfield_, Sep 03 2018