A090023 Number of distinct lines through the origin in the n-dimensional lattice of side length 7.

0, 1, 37, 415, 3745, 31471, 257257, 2078455, 16704865, 133935391, 1072633177, 8585561095, 68702163985, 549687102511, 4397773276297, 35183283965335, 281470638631105, 2251782504544831, 18014329402322617, 144114912035163175, 1152920401607386225

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) = 37 because in 2D the lines have slope 0, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 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

• Index entries for linear recurrences with constant coefficients, signature (18,-115,330,-424,192).

Crossrefs

Equals A090030(n+7,n).

Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090024 are for dimension n with side lengths 1, 2, 3, 4, 5, 6, 8 respectively. A049691, A090025, A090026, A090027, A090028, A090029 are for side length k in 2, 3, 4, 5, 6, 7 dimensions.

Programs

Mathematica

Table[8^n - 4^n - 3^n - 2^n + 2, {n, 0, 20}]

Python

[8**n-4**n-3**n-2**n+2 for n in range(25)] # Gennady Eremin, Mar 09 2022

Formula

a(n) = 8^n - 4^n - 3^n - 2^n + 2.

G.f.: -x*(200*x^3-136*x^2+19*x+1)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(8*x-1)). - Colin Barker, Sep 04 2012