A090022 Number of distinct lines through the origin in the n-dimensional lattice of side length 6.

0, 1, 25, 253, 2065, 15541, 112825, 804973, 5692705, 40071781, 281367625, 1972955293, 13823978545, 96820307221, 677949854425, 4746473419213, 33228592555585, 232613204977861, 1628344491013225, 11398619145204733

Offset: 0

Author

Joshua Zucker, Nov 20 2003

Keywords

• easy
• nonn

Comments

Equivalently, lattice points where the gcd of all the coordinates is 1.

Examples

			a(2) = 25 because in 2D the lines have slope 0, 1/6, 5/6, 1/5, 2/5, 3/5, 4/5, 1/4, 3/4, 1/3, 2/3, 1/2, 1 and their reciprocals.
		

Links

• Gennady Eremin, Table of n, a(n) for n = 0..500

Crossrefs

a(n) = T(n, 5) from A090030. Cf. A000225, A001047, A060867, A090020, A090021, A090023, A090024 are for dimension n with side lengths 1, 2, 3, 4, 5, 7, 8 respectively. A049691, A090025, A090026, A090027, A090028, A090029 are for side length k in 2, 3, 4, 5, 6, 7 dimensions.

Programs

Magma

[7^n-4^n-3^n+1: n in [0..20]]; // Wesley Ivan Hurt, Mar 06 2022

Mathematica

Table[7^n - 4^n - 3^n + 1, {n, 0, 25}]

Python

[7**n-4**n-3**n+1 for n in range(20)] # Gennady Eremin, Mar 06 2022

Formula

a(n) = 7^n - 4^n - 3^n + 1.

O.g.f.: 1/(-1+3*x) + 1/(-1+4*x) - 1/(-1+x) - 1/(-1+7*x). - R. J. Mathar, Feb 26 2008