A039685 Numbers m such that m^2 ends in 444.

38, 462, 538, 962, 1038, 1462, 1538, 1962, 2038, 2462, 2538, 2962, 3038, 3462, 3538, 3962, 4038, 4462, 4538, 4962, 5038, 5462, 5538, 5962, 6038, 6462, 6538, 6962, 7038, 7462, 7538, 7962, 8038, 8462, 8538, 8962, 9038, 9462, 9538, 9962, 10038, 10462

Offset

1,1

Comments

No square can end in more than three 4's.

When a square ends in exactly three identical digits, these digits are necessarily 444. - Bernard Schott, Oct 31 2019

References

Albert H. Beiler, "Recreations in the Theory of Numbers", Dover Publ., 2nd Ed. 1966, Chapter XV, "On The Square", p. 139. ISBN 0-486-21096-0.

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 1 pp. 55 and 95-96 (1995)

David Wells, "Curious and Interesting Numbers", Revised Ed. Penguin Books, p. 152. ISBN 0-14-026149-4.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

British Mathematical Olympiad, 1995 - Problem 1

Index entries for linear recurrences with constant coefficients, signature (1,1,-1). [From Bruno Berselli, Oct 27 2010]

Formula

a(2n+1) = 500n + 38 and a(2n+2) = 500n - 38.

From Bruno Berselli, Oct 27 2010: (Start)

a(n) = 250*n + 87*(-1)^n - 125.

G.f.: 2*x*(19 + 212*x + 19*x^2)/((1+x)*(1-x)^2).

a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3. (End)

Mathematica

Drop[ Flatten[ Table[{500n-38, 500n+38}, {n, 0, 21}]], 1] (* Robert G. Wilson v, Nov 27 2004 *)
Sqrt[#]&/@Select[Range[15000]^2, Mod[#, 1000]==444&] (* or *) LinearRecurrence[{1, 1, -1}, {38, 462, 538}, 50] (* Harvey P. Dale, Dec 26 2023 *)

Crossrefs

Cf. A328886 (squares that end in 444).

Sequence in context: A267474 A240258 A254471 * A006418 A160317 A088891

Adjacent sequences: A039682 A039683 A039684 * A039686 A039687 A039688

Keyword

nonn,base

Author

Felice Russo

Extensions

More terms from Patrick De Geest, Jun 15 1999

Status

approved