A090021 Number of distinct lines through the origin in the n-dimensional lattice of side length 5.

0, 1, 21, 175, 1185, 7471, 45801, 277495, 1672545, 10056991, 60405081, 362615815, 2176242705, 13059083311, 78359348361, 470170570135, 2821066729665, 16926530042431, 101559568723641, 609358576700455, 3656154951181425

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 (12,-47,72,-36).

Formula

a(n) = 6^n - 3^n - 2*2^n + 2.

G.f.: -x*(30*x^2-9*x-1)/((x-1)*(2*x-1)*(3*x-1)*(6*x-1)). [Colin Barker, Sep 04 2012]

Example

a(2) = 21 because in 2D the lines have slope 0, 1/5, 2/5, 3/5, 4/5, 1/4, 3/4, 1/3, 2/3, 1/2, 1 and their reciprocals.

Mathematica

Table[6^n - 3^n - 2*2^n + 2, {n, 0, 25}]
LinearRecurrence[{12, -47, 72, -36}, {0, 1, 21, 175}, 30] (* Harvey P. Dale, Jul 18 2016 *)

Crossrefs

a(n) = T(n, 5) from A090030. Cf. A000225, A001047, A060867, A090020, A090022, A090023, A090024 are for dimension n with side lengths 1, 2, 3, 4, 6, 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: A119105 A015880 A113163 * A254681 A219625 A244875

Adjacent sequences: A090018 A090019 A090020 * A090022 A090023 A090024

Keyword

easy,nonn

Author

Joshua Zucker, Nov 19 2003

Status

approved