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

3,1

Links

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

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

Example

     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.

Prog

(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

Crossrefs

Cf. A225606 (analogous problem for square grids).

Sequence in context: A203292 A054578 A115855 * A294387 A128128 A162920

Adjacent sequences: A234245 A234246 A234247 * A234249 A234250 A234251

Keyword

nonn,nice

Author

Heinrich Ludwig, Jan 18 2014

Extensions

More terms from Jon E. Schoenfield, Aug 17 2014

Status

approved