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A090025
Number of distinct lines through the origin in 3-dimensional cube of side length n.
14
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
OFFSET
0,2
COMMENTS
Equivalently, lattice points where the GCD of all coordinates = 1.
FORMULA
a(n) = A090030(3, n).
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
a(n) = 6*A015631(n) + 1 for n>=1. - Hugo Pfoertner, Mar 30 2021
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.
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}]
a[n_] := Sum[MoebiusMu[k]*((Floor[n/k]+1)^3-1), {k, 1, n}]; Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Nov 28 2013, after Vladeta Jovovic *)
PROG
(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)
def A090025(n):
if n == 0:
return 0
c, j = 1, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c += (j2-j)*A090025(k1)
j, k1 = j2, n//j2
return (n+1)**3-c+7*(j-n-1) # Chai Wah Wu, Mar 30 2021
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: A097039 A067651 A357301 * A348472 A003232 A018728
KEYWORD
nonn
AUTHOR
Joshua Zucker, Nov 25 2003
STATUS
approved