A090029 Number of distinct lines through the origin in 7-dimensional cube of side length n.

0, 127, 2059, 16129, 75811, 277495, 804973, 2078455, 4702531, 9905365, 19188793, 35533303, 61846723, 104511583, 168681913, 266042113, 405259513, 607140745, 883046011, 1269174145, 1780715833, 2472697501, 3366818491, 4548464341

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = A090030(7, n).

a(n) = (n+1)^7 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

Example

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.

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[7, k], {k, 0, 40}]

Prog

(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A090029(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A090029(k1)
        j, k1 = j2, n//j2
    return (n+1)**7-c+127*(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.

Sequence in context: A228263 A228222 A022523 * A152726 A069092 A024005

Adjacent sequences: A090026 A090027 A090028 * A090030 A090031 A090032

Keyword

nonn

Author

Joshua Zucker, Nov 25 2003

Status

approved