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A090023
Number of distinct lines through the origin in the n-dimensional lattice of side length 7.
11
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.
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
KEYWORD
easy,nonn
AUTHOR
Joshua Zucker, Nov 20 2003
STATUS
approved