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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equivalently, lattice points where the GCD of all coordinates = 1.

LINKS

Table of n, a(n) for n=0..39.

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-Fran├žois 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: A000491 A097039 A067651 * A003232 A018728 A027523

Adjacent sequences:  A090022 A090023 A090024 * A090026 A090027 A090028

KEYWORD

nonn

AUTHOR

Joshua Zucker, Nov 25 2003

STATUS

approved

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Last modified June 19 10:45 EDT 2021. Contains 345126 sequences. (Running on oeis4.)