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A090029
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Number of distinct lines through the origin in 7-dimensional cube of side length n.
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12
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0, 127, 2059, 16129, 75811, 277495, 804973, 2078455, 4702531, 9905365, 19188793, 35533303, 61846723, 104511583, 168681913, 266042113, 405259513, 607140745, 883046011, 1269174145, 1780715833, 2472697501, 3366818491, 4548464341
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OFFSET
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0,2
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COMMENTS
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Equivalently, lattice points where the GCD of all coordinates = 1.
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LINKS
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FORMULA
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a(n) = (n+1)^7 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021
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EXAMPLE
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a(2) = 2059 because the 2059 points with at least one coordinate=2 all make distinct lines and the remaining 127 points and the origin are on those lines.
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MATHEMATICA
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aux[n_, k_] := If[k == 0, 0, (k + 1)^n - k^n - Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]] - 1}]]; lines[n_, k_] := (k + 1)^n - Sum[Floor[k/i - 1]*aux[n, i], {i, 1, Floor[k/2]}] - 1; Table[lines[7, k], {k, 0, 40}]
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PROG
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(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
if n == 0:
return 0
c, j = 1, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
j, k1 = j2, n//j2
return (n+1)**7-c+127*(j-n-1) # Chai Wah Wu, Mar 30 2021
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CROSSREFS
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Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023, A090024 are for n dimensions with side length 1, 2, 3, 4, 5, 6, 7, 8, respectively. A049691, A090025, A090026, A090027, A090028, A090029 are this sequence for 2, 3, 4, 5, 6, 7 dimensions. A090030 is the table for n dimensions, side length k.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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