A090020 Number of distinct lines through the origin in the n-dimensional lattice of side length 4.

0, 1, 13, 91, 529, 2851, 14833, 75811, 383809, 1932931, 9705553, 48648931, 243605089, 1219100611, 6098716273, 30503196451, 152544778369, 762810181891, 3814309582993, 19072323542371, 95363943807649, 476826695752771

Offset: 0

Author

Joshua Zucker, Nov 19 2003

Keywords

• easy
• nonn

Comments

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

Examples

			a(2) = 13 because in 2D the lines have slope 0, 1/4, 1/3, 1/2, 2/3, 3/4, 1, 4/3, 3/2, 2, 3, 4 and infinity.
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (11,-41,61,-30).

Crossrefs

a(n) = T(n,4) from A090030. Cf. A000225, A001047, A060867, A090021, A090022, A090023, A090024 are for dimension n with side lengths 1, 2, 3, 5, 6, 7, 8 respectively. A049691, A090025, A090026, A090027, A090028, A090029 are for side length k in 2, 3, 4, 5, 6, 7 dimensions.

Programs

Mathematica

Table[5^n - 3^n - 2^n + 1, {n, 0, 25}]
LinearRecurrence[{11,-41,61,-30},{0,1,13,91},30] (* Indranil Ghosh, Feb 21 2017 *)

Python

def A090020(n): return 5**n-3**n-2**n+1 # Indranil Ghosh, Feb 21 2017

Formula

a(n) = 5^n - 3^n - 2^n + 1.

G.f.: -x*(11*x^2-2*x-1)/((x-1)*(2*x-1)*(3*x-1)*(5*x-1)). [Colin Barker, Sep 04 2012]