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A090026 Number of distinct lines through the origin in 4-dimensional cube of side length n.

%I A090026 #13 Mar 30 2021 21:09:13
%S A090026 0,15,65,225,529,1185,2065,3745,5841,9105,13025,19105,25521,35361,
%T A090026 45825,59905,75425,96865,117841,147505,177041,214961,254401,306321,
%U A090026 355249,420929,485489,565265,645377,748081,841841,966881,1086241,1230401,1373185,1549825
%N A090026 Number of distinct lines through the origin in 4-dimensional cube of side length n.
%C A090026 Equivalently, number of lattice points where the GCD of all coordinates = 1.
%F A090026 a(n) = A090030(4, n).
%F A090026 a(n) = (n+1)^4 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - _Chai Wah Wu_, Mar 30 2021
%e A090026 a(2) = 65 because the 65 points with at least one coordinate=2 all make distinct lines and the remaining 15 points and the origin are on those lines.
%t A090026 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[4, k], {k, 0, 40}]
%o A090026 (Python)
%o A090026 from functools import lru_cache
%o A090026 @lru_cache(maxsize=None)
%o A090026 def A090026(n):
%o A090026     if n == 0:
%o A090026         return 0
%o A090026     c, j = 1, 2
%o A090026     k1 = n//j
%o A090026     while k1 > 1:
%o A090026         j2 = n//k1 + 1
%o A090026         c += (j2-j)*A090026(k1)
%o A090026         j, k1 = j2, n//j2
%o A090026     return (n+1)**4-c+15*(j-n-1) # _Chai Wah Wu_, Mar 30 2021
%Y A090026 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 A090026 nonn
%O A090026 0,2
%A A090026 _Joshua Zucker_, Nov 25 2003