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A090021
Number of distinct lines through the origin in the n-dimensional lattice of side length 5.
11
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.
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
KEYWORD
easy,nonn
AUTHOR
Joshua Zucker, Nov 19 2003
STATUS
approved