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A090028 Number of distinct lines through the origin in 6-dimensional cube of side length n.

%I A090028 #16 Mar 31 2021 09:45:29
%S A090028 0,63,665,3969,14833,45801,112825,257257,515025,980217,1720145,
%T A090028 2934505,4693473,7396137,11112129,16464385,23555441,33430033,45927505,
%U A090028 62881561,83865257,111331241,144772201,187839225,238778281,303522401,379323785
%N A090028 Number of distinct lines through the origin in 6-dimensional cube of side length n.
%C A090028 Equivalently, lattice points where the GCD of all coordinates = 1.
%F A090028 a(n) = A090030(6, n).
%F A090028 a(n) = (n+1)^6 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - _Chai Wah Wu_, Mar 30 2021
%e A090028 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.
%t A090028 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}]
%o A090028 (Python)
%o A090028 from functools import lru_cache
%o A090028 @lru_cache(maxsize=None)
%o A090028 def A090028(n):
%o A090028     if n == 0:
%o A090028         return 0
%o A090028     c, j = 1, 2
%o A090028     k1 = n//j
%o A090028     while k1 > 1:
%o A090028         j2 = n//k1 + 1
%o A090028         c += (j2-j)*A090028(k1)
%o A090028         j, k1 = j2, n//j2
%o A090028     return (n+1)**6-c+63*(j-n-1) # _Chai Wah Wu_, Mar 30 2021
%Y A090028 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.
%K A090028 nonn
%O A090028 0,2
%A A090028 _Joshua Zucker_, Nov 25 2003