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A090025
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Number of distinct lines through the origin in 3-dimensional cube of side length n.
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13
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0, 7, 19, 49, 91, 175, 253, 415, 571, 805, 1033, 1423, 1723, 2263, 2713, 3313, 3913, 4825, 5491, 6625, 7513, 8701, 9811, 11461, 12637, 14497, 16045, 18043, 19807, 22411, 24163, 27133, 29485, 32425, 35065, 38593, 41221, 45433, 48727, 52831
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Equivalently, lattice points where the GCD of all coordinates = 1.
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FORMULA
| a(n) = A090030(3, n)
a(n) = Sum(moebius(k)*((floor(n/k)+1)^3-1), k=1..n). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 03 2004
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EXAMPLE
| a(2) = 19 because the 19 points with at least one coordinate=2 all make distinct lines and the remaining 7 points and the origin are on those lines.
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MATHEMATICA
| 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[3, k], {k, 0, 40}]
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CROSSREFS
| 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.
Cf. A071778.
Sequence in context: A000491 A097039 A067651 * A003232 A018728 A027523
Adjacent sequences: A090022 A090023 A090024 * A090026 A090027 A090028
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KEYWORD
| nonn
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AUTHOR
| Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Nov 25 2003
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