A244504 Number of distinct lines passing through at least two points in a triangular grid of side n.

3, 9, 24, 51, 102, 177, 294, 459, 690, 987, 1380, 1875, 2508, 3279, 4212, 5319, 6648, 8199, 10026, 12141, 14580, 17343, 20496, 24051, 28068, 32547, 37542, 43071, 49218, 55983, 63456, 71661, 80658, 90447, 101100, 112635, 125160, 138675, 153252, 168915, 185784

Offset

2,1

Links

Heinrich Ludwig, Table of n, a(n) for n = 2..1000

Valentina Beorchia, Matteo Gallet, and Alessandro Logar, Free plane curves with a linear Jacobian syzygy, arXiv:2512.10928 [math.AC], 2025. See p. 12.

Formula

a(n) = 3*Sum_{j = 1..n-1} euler_phi(j)*(g(n-j)-g(n-2*j)), where g(i) = i*(i+1)/2 if i > 0, otherwise 0, after Jon E. Schoenfield.

Mathematica

g[i_]:=If[i>0, i*(i+1)/2, 0]; Table[3*Sum[EulerPhi[j]*(g[n-j]-g[n-2*j]), {j, 1, n-1}], {n, 2, 50}] (* Vaclav Kotesovec, Sep 04 2014 after Jon E. Schoenfield *)

Prog

(PARI) g(j) = if (j > 0, j*(j+1)/2, 0);
a(n) = 3*sum(j = 1, n-1, eulerphi(j)*(g(n-j)-g(n-2*j))); \\ Michel Marcus, Sep 04 2014

Crossrefs

Cf. A234248.

Sequence in context: A029530 A301740 A227018 * A378854 A085739 A245762

Adjacent sequences: A244501 A244502 A244503 * A244505 A244506 A244507

Keyword

nonn

Author

Heinrich Ludwig, Sep 04 2014

Status

approved