A189416 Number of parallelograms on an n X n grid.

0, 1, 22, 158, 674, 2159, 5664, 13004, 26904, 51401, 92094, 156710, 255090, 400359, 608656, 900100, 1299336, 1836461, 2546550, 3472162, 4661898, 6173123, 8071952, 10434600, 13346080, 16905033, 21221558, 26419338, 32636098, 40027283, 48761448

Offset: 1

Author

Martin Renner, Apr 21 2011

Keywords

• nonn

Links

• Nathaniel Johnston, C program for computing terms
• Eric Weisstein's World of Mathematics, Parallelogram

Crossrefs

Cf. A186434, A189413, A189414.

Programs

Mathematica

a[n_] := Sum[(n-a)*(n-b)*(2*a*b - GCD[a, b]), {a, 1, n-1}, {b, 1, n-1}];
Array[a, 31] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

PARI

a(n) = sum(a=1, n-1, sum(b=1, n-1, (n-a)*(n-b)*(2*a*b - gcd(a,b)) )); \\ Andrew Howroyd, Sep 19 2017

Formula

a(n) = Sum_{a=1..n-1} Sum_{b=1..n-1} (n-a)*(n-b)*(2*a*b - gcd(a,b)). - Andrew Howroyd, Sep 19 2017

Extensions

a(6)-a(31) from Nathaniel Johnston, Apr 24 2011