A090030 Triangle read by rows: T(n,k) = number of distinct lines through the origin in the n-dimensional cubic lattice of side length k with one corner at the origin.
0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 1, 5, 7, 0, 0, 1, 9, 19, 15, 0, 0, 1, 13, 49, 65, 31, 0, 0, 1, 21, 91, 225, 211, 63, 0, 0, 1, 25, 175, 529, 961, 665, 127, 0, 0, 1, 37, 253, 1185, 2851, 3969, 2059, 255, 0, 0, 1, 45, 415, 2065, 7471, 14833, 16129, 6305, 511, 0, 0, 1, 57, 571, 3745, 15541, 45801, 75811, 65025, 19171, 1023, 0
Offset: 0
Examples
T(n,1) = 2^n-1 because there are 2^n-1 lattice points other than the corner, all of which make distinct lines. T(n,2) = 3^n - 2^n because if the given corner is the origin, all the points with coordinates in {0,1} make lines that are redundant with a point containing a coordinate 2.
Triangle T(n,k) begins:
0;
0, 0;
0, 1, 0;
0, 1, 3, 0;
0, 1, 5, 7, 0;
0, 1, 9, 19, 15, 0;
0, 1, 13, 49, 65, 31, 0;
0, 1, 21, 91, 225, 211, 63, 0;
0, 1, 25, 175, 529, 961, 665, 127, 0;
0, 1, 37, 253, 1185, 2851, 3969, 2059, 255, 0;
0, 1, 45, 415, 2065, 7471, 14833, 16129, 6305, 511, 0;
...
Crossrefs
Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023, A090024 give T(n, k) for k = 1, 2, 3, 4, 5, 6, 7, 8, respectively. A049691, A090025, A090026, A090027, A090028, A090029 give T(n, k) for n=2, 3, 4, 5, 6, 7 respectively. A090225 counts only points with at least one coordinate = k.
T(n,n) gives A081474.
Cf. A008683.
Programs
-
Mathematica
aux[n_, k_] := If[k==0, 0, (k+1)^n-k^n-Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]]-1}]];lines[n_, k_] := (k+1)^n-Sum[Floor[k/i-1]*aux[n, i], {i, 1, Floor[k/2]}]-1;lines[n, k]
Formula
With A(n, k) = A090225(n, k), T(n, k) =(k+1)^n - 1 - the sum for 0 < i < k of Floor[k/i-1]*A(n, i)
T(n,k) = Sum_{i=1..n-k} moebius(i)*((floor((n-k)/i)+1)^k-1). - Vladeta Jovovic, Dec 03 2004
Comments