A090030 -id:A090030 - cp's OEIS frontend

A081474 Number of distinct lines through the origin in n-dimensional cube of side length n.

0, 1, 5, 49, 529, 7471, 112825, 2078455, 42649281, 997784221, 25875851825, 742641202183, 23283999690561, 793616663524231, 29188521870580929, 1152885848976064513, 48659336030073207425, 2185894865613157551481, 104126348669497256201905, 5242869988601103651841105

Offset: 0

Joshua Zucker, Nov 25 2003

nonn

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

			a(3) = 49 because in the 3-dimensional lattice of side length 3, the lines through the origin are determined by all 37 points with at least one coordinate = 3 and 6 permutations of (2,1,0) and 3 permutations each of (2,1,1) and (2,2,1).

Alois P. Heinz, Table of n, a(n) for n = 0..386

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.

a:= n-> add(numtheory[mobius](i)*((floor(n/i)+1)^n-1), i=1..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 09 2022

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[k, k], {k, 0, 20}]

a(n) = A090030(n,n).

A090225 T(n,k) = Points in n-dimensional lattice of side length k with at least one coordinate = k and GCD of all coordinates = 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 7, 0, 0, 0, 4, 12, 15, 0, 0, 0, 4, 30, 50, 31, 0, 0, 0, 8, 42, 160, 180, 63, 0, 0, 0, 4, 84, 304, 750, 602, 127, 0, 0, 0, 12, 78, 656, 1890, 3304, 1932, 255, 0, 0, 0, 8, 162, 880, 4620, 10864, 14070, 6050, 511, 0, 0, 0, 12, 156, 1680, 8070

Offset: 0

Joshua Zucker, Nov 24 2003

			T(3,2) = 12 because of the six permutations of (2,1,0) and three each of (2,1,1) and (2,2,1).
0;
0,  0;
0,  1,  0;
0,  0,  3,  0;
0,  0,  2,  7,  0;
0,  0,  4, 12, 15, 0;
0,  0,  4, 30, 50, 31,  0;
0,  0,  8, 42,160,180, 63, 0;
0,  0,  4, 84,304,750,602,127, 0;
0,  0, 12, 78,656,1890,3304,1932,255, 0;

Cf. A090030.

aux[n_, k_] := If[k == 0, 0, (k + 1)^n - k^n - Sum[aux[n, Divisors[k][[i]]], {i, 1, Length[Divisors[k]] - 1}]]

T(n, 0) = 0; T(n, k) = (k+1)^n - k^n - sum T(n, divisors of k)

cp's OEIS Frontend

A081474 Number of distinct lines through the origin in n-dimensional cube of side length n.

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