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A090025 Number of distinct lines through the origin in 3-dimensional cube of side length n. 14

%I #26 Mar 30 2021 18:44:17

%S 0,7,19,49,91,175,253,415,571,805,1033,1423,1723,2263,2713,3313,3913,

%T 4825,5491,6625,7513,8701,9811,11461,12637,14497,16045,18043,19807,

%U 22411,24163,27133,29485,32425,35065,38593,41221,45433,48727,52831

%N Number of distinct lines through the origin in 3-dimensional cube of side length n.

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

%F a(n) = A090030(3, n).

%F a(n) = Sum_{k=1..n} moebius(k)*((floor(n/k)+1)^3-1). - _Vladeta Jovovic_, Dec 03 2004

%F a(n) = (n+1)^3 - Sum_{j=2..n+1} a(floor(n/j)). - _Seth A. Troisi_, Aug 29 2013

%F a(n) = 6*A015631(n) + 1 for n>=1. - _Hugo Pfoertner_, Mar 30 2021

%e a(2) = 19 because the 19 points with at least one coordinate=2 all make distinct lines and the remaining 7 points and the origin are on those lines.

%t 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[3, k], {k, 0, 40}]

%t a[n_] := Sum[MoebiusMu[k]*((Floor[n/k]+1)^3-1), {k, 1, n}]; Table[a[n], {n, 0, 39}] (* _Jean-François Alcover_, Nov 28 2013, after _Vladeta Jovovic_ *)

%o (PARI) a(n)=(n+1)^3-sum(j=2,n+1,a(floor(n/j)))

%o (Python)

%o from functools import lru_cache

%o @lru_cache(maxsize=None)

%o def A090025(n):

%o if n == 0:

%o return 0

%o c, j = 1, 2

%o k1 = n//j

%o while k1 > 1:

%o j2 = n//k1 + 1

%o c += (j2-j)*A090025(k1)

%o j, k1 = j2, n//j2

%o return (n+1)**3-c+7*(j-n-1) # _Chai Wah Wu_, Mar 30 2021

%Y 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.

%Y Cf. A071778.

%K nonn

%O 0,2

%A _Joshua Zucker_, Nov 25 2003

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)