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A039685
Numbers m such that m^2 ends in 444.
10
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
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)
E.g.f.: 38 + (250*x - 38)*cosh(x) + (250*x - 212)*sinh(x). - Stefano Spezia, Sep 15 2024
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
KEYWORD
nonn,base,easy
AUTHOR
EXTENSIONS
More terms from Patrick De Geest, Jun 15 1999
STATUS
approved