Monatshefte für Mathematik

, Volume 164, Issue 2, pp 157–170 | Cite as

Formulas for the number of gridlines

  • Anne-Maria Ernvall-Hytönen
  • Kaisa MatomäkiEmail author
  • Pentti Haukkanen
  • Jorma K. Merikoski


Let l(n) be the number of lines through at least two points of an n × n rectangular grid. We prove recursive and asymptotic formulas for it using respectively combinatorial and number theoretic methods. We also study the ratio l(n)/l(n − 1). All this originates from Mustonen’s experimental results.


Rectangular grid Lattice points Euler \({\phi}\)-function Asymptotic formulas Recursive formulas 

Mathematics Subject Classification (2000)

05A99 11B37 11N37 11P21 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Anne-Maria Ernvall-Hytönen
    • 1
  • Kaisa Matomäki
    • 2
    Email author
  • Pentti Haukkanen
    • 3
  • Jorma K. Merikoski
    • 3
  1. 1.Institutionen för matematik, Kungliga Tekniska HögskolanStockholmSweden
  2. 2.Department of MathematicsUniversity of TurkuTurkuFinland
  3. 3.Department of Mathematics and StatisticsUniversity of TampereTampereFinland

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