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).
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
KEYWORD
nonn,base,easy
AUTHOR
EXTENSIONS
More terms from Patrick De Geest, Jun 15 1999
STATUS
approved