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

0,3

Comments

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

Links

Table of n, a(n) for n=0..20.

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

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

Example

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.

Mathematica

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

Prog

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

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.

Sequence in context: A201789 A115926 A083818 * A254682 A232251 A232259

Adjacent sequences: A090020 A090021 A090022 * A090024 A090025 A090026

Keyword

easy,nonn

Author

Joshua Zucker, Nov 20 2003

Status

approved