A234248 Number of distinct lines passing through at least three points in a triangular grid of side n.

3, 6, 12, 21, 36, 57, 90, 129, 186, 261, 354, 465, 612, 783, 990, 1233, 1524, 1863, 2262, 2703, 3216, 3801, 4458, 5187, 6024, 6951, 7986, 9129, 10392, 11775, 13302, 14943, 16746, 18711, 20844, 23145, 25668, 28377, 31296, 34425, 37782, 41367, 45210, 49287

Offset: 3

Author

Heinrich Ludwig, Jan 18 2014

Keywords

• nonn
• nice

Examples

			     a
    b c
   d e f
  g h i j
In this triangle grid of side 4, there are a(4) = 6 distinct lines passing through at least 3 points: ag, gj, ja, ch, df, ib.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 3..10000

Crossrefs

Cf. A225606 (analogous problem for square grids).

Programs

PARI

g(n) = if (n>0, n*(n+1)/2, 0);
a(n) = my(k=3); 3*sum(j=1, (n-1)\(k-1), eulerphi(j) * (g(n-(k-1)*j) - g(n-k*j))); \\ Michel Marcus, Aug 19 2014

Formula

a(n) = 3*Sum_{j=1..floor((n-1)/(k-1))} EulerPhi(j) * (g(n-(k-1)*j) - g(n-k*j)) where k = 3 (the minimum required number of points) and g(i) = A000217(i) (i.e., the i-th triangular number) if i > 0, otherwise 0. - Jon E. Schoenfield, Aug 17 2014

Extensions

More terms from Jon E. Schoenfield, Aug 17 2014