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A350382 a(n) = 9 + 4 * 10^n.

%I A350382 #37 Jan 11 2022 22:07:43
%S A350382 49,409,4009,40009,400009,4000009,40000009,400000009,4000000009,
%T A350382 40000000009,400000000009,4000000000009,40000000000009,
%U A350382 400000000000009,4000000000000009,40000000000000009,400000000000000009,4000000000000000009,40000000000000000009,400000000000000000009,4000000000000000000009
%N A350382 a(n) = 9 + 4 * 10^n.
%C A350382 The 4th problem of 16th Tournament of Towns in 1994-1995, Spring tour 1995, 8-9 grades, Training option, asked for a proof that the number 400...009 with at least one zero is not a perfect square (see link).
%C A350382 Indeed, the first few squares whose digits are 0, 4 and 9 are 4900, 9409, 490000, 940900, 994009, ... (comes from A019544).
%C A350382 Generalization: the 4th problem of 16th Tournament of Towns in 1994-1995, Spring tour 1995, 10-11 grades, Training option, asked for a proof that the number d00...009 with at least one zero is not a perfect square, when d is a digit with 1 <= d <= 9 (see link).
%D A350382 Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 (in fact, it is Problem 4) of Tournament of Towns 1995, page 301.
%H A350382 Tournament of Towns 1994-1995, Spring tour <a href="/A350382/a350382.pdf">Problem 4, 8-9 grades, Training option & Problem 4, 10-11 grades, Training option</a> (in Russian and English, problems in red).
%H A350382 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-10).
%H A350382 <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.
%F A350382 a(n) = 9 + 4*10^n = 4*A133384(n-1) + 1.
%F A350382 a(n) = 24*A126109(n-1) + 1 = 10*A199684(n-1) - 1. - _Hugo Pfoertner_, Dec 28 2021
%F A350382 From _Stefano Spezia_, Dec 28 2021: (Start)
%F A350382 G.f.: x*(49 - 130*x)/((1 - x)*(1 - 10*x)).
%F A350382 a(n) = 11*a(n-1) - 10*a(n-2) for n > 2. (End)
%e A350382 a(3) = 9 + 4 * 10^3 = 4009 = 19 * 211 is not a square.
%p A350382 Data := [seq(9 + 4*10^n,  n = 1..20)];
%t A350382 a[n_] := 9 + 4*10^n; Array[a, 20] (* _Amiram Eldar_, Dec 28 2021 *)
%Y A350382 Cf. A019544, A126109, A133384, A199684.
%K A350382 nonn,base,easy
%O A350382 1,1
%A A350382 _Bernard Schott_, Dec 28 2021