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A350382
a(n) = 9 + 4 * 10^n.
1
49, 409, 4009, 40009, 400009, 4000009, 40000009, 400000009, 4000000009, 40000000009, 400000000009, 4000000000009, 40000000000009, 400000000000009, 4000000000000009, 40000000000000009, 400000000000000009, 4000000000000000009, 40000000000000000009, 400000000000000000009, 4000000000000000000009
OFFSET
1,1
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).
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.
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)
EXAMPLE
a(3) = 9 + 4 * 10^3 = 4009 = 19 * 211 is not a square.
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 *)
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Bernard Schott, Dec 28 2021
STATUS
approved