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

49, 409, 4009, 40009, 400009, 4000009, 40000009, 400000009, 4000000009, 40000000009, 400000000009, 4000000000009, 40000000000009, 400000000000009, 4000000000000009, 40000000000000009, 400000000000000009, 4000000000000000009, 40000000000000000009, 400000000000000000009, 4000000000000000000009

Offset: 1

Author

Bernard Schott, Dec 28 2021

Keywords

• nonn
• base
• easy

Comments

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).

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

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).

Examples

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

References

• 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.

Links

• Tournament of Towns 1994-1995, Spring tour Problem 4, 8-9 grades, Training option & Problem 4, 10-11 grades, Training option (in Russian and English, problems in red).
• Index entries for linear recurrences with constant coefficients, signature (11,-10).
• Index to sequences related to Olympiads and other Mathematical competitions.

Crossrefs

Cf. A019544, A126109, A133384, A199684.

Programs

Maple

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

Mathematica

a[n_] := 9 + 4*10^n; Array[a, 20] (* Amiram Eldar, Dec 28 2021 *)

Formula

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

a(n) = 24*A126109(n-1) + 1 = 10*A199684(n-1) - 1. - Hugo Pfoertner, Dec 28 2021

From Stefano Spezia, Dec 28 2021: (Start)

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

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