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 A090026 Number of distinct lines through the origin in 4-dimensional cube of side length n. 12
 0, 15, 65, 225, 529, 1185, 2065, 3745, 5841, 9105, 13025, 19105, 25521, 35361, 45825, 59905, 75425, 96865, 117841, 147505, 177041, 214961, 254401, 306321, 355249, 420929, 485489, 565265, 645377, 748081, 841841, 966881, 1086241, 1230401, 1373185, 1549825 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Equivalently, number of lattice points where the GCD of all coordinates = 1. LINKS FORMULA a(n) = A090030(4, n). a(n) = (n+1)^4 - 1 - Sum_{j=2..n+1} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021 EXAMPLE 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. 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[4, k], {k, 0, 40}] PROG (Python) from functools import lru_cache @lru_cache(maxsize=None) def A090026(n):     if n == 0:         return 0     c, j = 1, 2     k1 = n//j     while k1 > 1:         j2 = n//k1 + 1         c += (j2-j)*A090026(k1)         j, k1 = j2, n//j2     return (n+1)**4-c+15*(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: A027455 A152729 A055268 * A027526 A334802 A284898 Adjacent sequences:  A090023 A090024 A090025 * A090027 A090028 A090029 KEYWORD nonn AUTHOR Joshua Zucker, Nov 25 2003 STATUS approved

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Last modified June 15 19:31 EDT 2021. Contains 345049 sequences. (Running on oeis4.)