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a(n) = 9 + 4 * 10^n.
1

%I #37 Jan 11 2022 22:07:43

%S 49,409,4009,40009,400009,4000009,40000009,400000009,4000000009,

%T 40000000009,400000000009,4000000000009,40000000000009,

%U 400000000000009,4000000000000009,40000000000000009,400000000000000009,4000000000000000009,40000000000000000009,400000000000000000009,4000000000000000000009

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

%C 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 Indeed, the first few squares whose digits are 0, 4 and 9 are 4900, 9409, 490000, 940900, 994009, ... (comes from A019544).

%C 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 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 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-10).

%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.

%F a(n) = 9 + 4*10^n = 4*A133384(n-1) + 1.

%F a(n) = 24*A126109(n-1) + 1 = 10*A199684(n-1) - 1. - _Hugo Pfoertner_, Dec 28 2021

%F From _Stefano Spezia_, Dec 28 2021: (Start)

%F G.f.: x*(49 - 130*x)/((1 - x)*(1 - 10*x)).

%F a(n) = 11*a(n-1) - 10*a(n-2) for n > 2. (End)

%e a(3) = 9 + 4 * 10^3 = 4009 = 19 * 211 is not a square.

%p Data := [seq(9 + 4*10^n, n = 1..20)];

%t a[n_] := 9 + 4*10^n; Array[a, 20] (* _Amiram Eldar_, Dec 28 2021 *)

%Y Cf. A019544, A126109, A133384, A199684.

%K nonn,base,easy

%O 1,1

%A _Bernard Schott_, Dec 28 2021