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A090020
Number of distinct lines through the origin in the n-dimensional lattice of side length 4.
12
0, 1, 13, 91, 529, 2851, 14833, 75811, 383809, 1932931, 9705553, 48648931, 243605089, 1219100611, 6098716273, 30503196451, 152544778369, 762810181891, 3814309582993, 19072323542371, 95363943807649, 476826695752771
OFFSET
0,3
COMMENTS
Equivalently, lattice points where the gcd of all the coordinates is 1.
FORMULA
a(n) = 5^n - 3^n - 2^n + 1.
G.f.: -x*(11*x^2-2*x-1)/((x-1)*(2*x-1)*(3*x-1)*(5*x-1)). [Colin Barker, Sep 04 2012]
EXAMPLE
a(2) = 13 because in 2D the lines have slope 0, 1/4, 1/3, 1/2, 2/3, 3/4, 1, 4/3, 3/2, 2, 3, 4 and infinity.
MATHEMATICA
Table[5^n - 3^n - 2^n + 1, {n, 0, 25}]
LinearRecurrence[{11, -41, 61, -30}, {0, 1, 13, 91}, 30] (* Indranil Ghosh, Feb 21 2017 *)
PROG
(Python) def A090020(n): return 5**n-3**n-2**n+1 # Indranil Ghosh, Feb 21 2017
CROSSREFS
a(n) = T(n,4) from A090030. Cf. A000225, A001047, A060867, A090021, A090022, A090023, A090024 are for dimension n with side lengths 1, 2, 3, 5, 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: A010965 A221144 A022578 * A092469 A300779 A275918
KEYWORD
easy,nonn
AUTHOR
Joshua Zucker, Nov 19 2003
STATUS
approved