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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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14
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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
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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) = Sum_{k=1..n} moebius(k)*((floor(n/k)+1)^3-1). - Vladeta Jovovic, Dec 03 2004
a(n) = (n+1)^3 - Sum_{j=2..n+1} a(floor(n/j)). - Seth A. Troisi, Aug 29 2013
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EXAMPLE
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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
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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[3, k], {k, 0, 40}]
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PROG
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(PARI) a(n)=(n+1)^3-sum(j=2, n+1, a(floor(n/j)))
(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
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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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