A234248 - OEIS

%I #32 Jul 06 2019 15:27:22

%S 3,6,12,21,36,57,90,129,186,261,354,465,612,783,990,1233,1524,1863,

%T 2262,2703,3216,3801,4458,5187,6024,6951,7986,9129,10392,11775,13302,

%U 14943,16746,18711,20844,23145,25668,28377,31296,34425,37782,41367,45210,49287

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

%H Jon E. Schoenfield, <a href="/A234248/b234248.txt">Table of n, a(n) for n = 3..10000</a>

%F 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

%e      a

%e     b c

%e    d e f

%e   g h i j

%e 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.

%o (PARI) g(n) = if (n>0, n*(n+1)/2, 0);

%o 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

%Y Cf. A225606 (analogous problem for square grids).

%K nonn,nice

%O 3,1

%A _Heinrich Ludwig_, Jan 18 2014

%E More terms from _Jon E. Schoenfield_, Aug 17 2014