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

0,3

Comments

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

Links

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

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

Example

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.

Mathematica

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

Prog

(Python) [7**n-4**n-3**n+1 for n in range(20)] # Gennady Eremin, Mar 06 2022
(Magma) [7^n-4^n-3^n+1: n in [0..20]]; // Wesley Ivan Hurt, Mar 06 2022

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.

Sequence in context: A308492 A042208 A143009 * A298068 A017450 A264267

Adjacent sequences: A090019 A090020 A090021 * A090023 A090024 A090025

Keyword

easy,nonn

Author

Joshua Zucker, Nov 20 2003

Status

approved