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A090028 Number of distinct lines through the origin in 6-dimensional cube of side length n. 12
0, 63, 665, 3969, 14833, 45801, 112825, 257257, 515025, 980217, 1720145, 2934505, 4693473, 7396137, 11112129, 16464385, 23555441, 33430033, 45927505, 62881561, 83865257, 111331241, 144772201, 187839225, 238778281, 303522401, 379323785 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Equivalently, lattice points where the GCD of all coordinates = 1.
LINKS
FORMULA
a(n) = A090030(6, n).
a(n) = (n+1)^6 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021
EXAMPLE
a(2) = 665 because the 665 points with at least one coordinate=2 all make distinct lines and the remaining 63 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[6, k], {k, 0, 40}]
PROG
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A090028(n):
if n == 0:
return 0
c, j = 1, 2
k1 = n//j
while k1 > 1:
j2 = n//k1 + 1
c += (j2-j)*A090028(k1)
j, k1 = j2, n//j2
return (n+1)**6-c+63*(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: A228221 A022522 A152731 * A152725 A086578 A198399
KEYWORD
nonn
AUTHOR
Joshua Zucker, Nov 25 2003
STATUS
approved

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)