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A090024
Number of distinct lines through the origin in the n-dimensional lattice of side length 8.
12
0, 1, 45, 571, 5841, 55651, 515025, 4702531, 42649281, 385447171, 3476958705, 31332052291, 282184860321, 2540643522691, 22870684139985, 205860600134851, 1852867557848961, 16676418630942211, 150090820212050865
OFFSET
0,3
COMMENTS
Equivalently, lattice points where the gcd of all the coordinates is 1.
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]
EXAMPLE
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.
MATHEMATICA
Table[9^n - 5^n - 3^n - 2^n + 2, {n, 0, 20}]
PROG
(Python)
[9**n-5**n-3**n-2**n+2 for n in range(30)] # Gennady Eremin, Mar 12 2022
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.
Sequence in context: A160837 A160838 A189350 * A282767 A372978 A160234
KEYWORD
easy,nonn
AUTHOR
Joshua Zucker, Nov 20 2003
STATUS
approved