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

0,9

Comments

Equivalently, number of lattice points where the GCD of all coordinates = 1.

Links

Table of n, a(n) for n=0..77.

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

Example

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;
  ...

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]

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.

Sequence in context: A325735 A235794 A349911 * A293616 A211649 A202023

Adjacent sequences: A090027 A090028 A090029 * A090031 A090032 A090033

Keyword

nonn,tabl

Author

Joshua Zucker, Nov 24 2003

Status

approved