A090024 Number of distinct lines through the origin in the n-dimensional lattice of side length 8.

0, 1, 45, 571, 5841, 55651, 515025, 4702531, 42649281, 385447171, 3476958705, 31332052291, 282184860321, 2540643522691, 22870684139985, 205860600134851, 1852867557848961, 16676418630942211, 150090820212050865

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) = 45 because in 2D the lines have slope 0, 1/8, 3/8, 5/8, 7/8, 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

• Gennady Eremin, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (20,-140,430,-579,270).

Crossrefs

a(n) = T(n, 5) from A090030. Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023 are for dimension n with side lengths 1, 2, 3, 4, 5, 6, 7 respectively. A049691, A090025, A090026, A090027, A090028, A090029 are for side length k in 2, 3, 4, 5, 6, 7 dimensions.

Programs

Mathematica

Table[9^n - 5^n - 3^n - 2^n + 2, {n, 0, 20}]

Python

[9**n-5**n-3**n-2**n+2 for n in range(30)] # Gennady Eremin, Mar 12 2022

Formula

a(n) = 9^n - 5^n - 3^n - 2^n + 2.

G.f.: -x*(291*x^3-189*x^2+25*x+1)/((x-1)*(2*x-1)*(3*x-1)*(5*x-1)*(9*x-1)). [Colin Barker, Sep 04 2012]