A300778 -id:A300778 - cp's OEIS frontend

A049687 Array T read by diagonals: T(i,j)=number of lines passing through (0,0) and at least one other lattice point (h,k) satisfying 0<=h<=i, 0<=k<=j.

0, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 5, 5, 1, 1, 6, 7, 7, 6, 1, 1, 7, 8, 9, 8, 7, 1, 1, 8, 10, 11, 11, 10, 8, 1, 1, 9, 11, 14, 13, 14, 11, 9, 1, 1, 10, 13, 15, 17, 17, 15, 13, 10, 1, 1, 11, 14, 18, 18, 21, 18, 18, 14, 11, 1, 1, 12, 16, 20, 22, 23, 23, 22, 20, 16, 12, 1, 1, 13, 17, 22, 24

Offset: 0

Clark Kimberling

			The array begins:
0 1 1 1 1  ...
1 3 4 5 6  ...
1 4 5 7 8  ...
1 5 7 9 11 ...
1 6 8 11 13 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Main diagonal is A049691.

Cf. A049639, A135646, A300778.

a[0, 0] = 0; a[0, ] = a[, 0] = 1; a[i_, j_] := Module[{slopes, cnt}, slopes = Union @ Flatten @ Table[k/h, {h, 1, i }, {k, 1, j }]; cnt[ slope_] := Count[Flatten[Table[{h, k}, {h, 1, i }, {k, 1, j }], 1], {h_, k_} /; k/h == slope]; Count[cnt /@ slopes, c_ /; c >= 1] + 2]; Table[a[i-j, j], {i, 0, 12}, {j, 0, i}] // Flatten (* Jean-François Alcover, Apr 03 2017 *)

T(i,j) = (i>0) + (j>0) + sum(g=1, min(i,j), (i\g) * (j\g) * moebius(g));
for (i=0, 10, for(j=0, 10, print1(T(i,j), ", ")); print); \\ Andrew Howroyd, Sep 17 2017

T(i,j) = sum(h=0, i, sum(k=0, j, gcd(h,k) == 1)); \\ Andrew Howroyd, Sep 17 2017

From Andrew Howroyd, Sep 17 2017: (Start)
T(i, j) = 2 + Sum_{g=1..min(i,j)} floor(i/g) * floor(j/g) * mu(g) for i > 0, j > 0.
T(i, j) = signum(i) + signum(j) + A135646(i, j).
T(i, j) = |{(x, y): gcd(x, y) = 1, 0<=x<=i, 0<=y<=j}|.
(End)

More terms from Michael Somos

A339400 Mark each point on the n X n grid with the number of points that are visible from it; a(n) is the number of distinct values in the grid.

Original entry on oeis.org

1, 3, 3, 4, 3, 7, 5, 7, 7, 11, 5, 14, 8, 13, 13, 19, 9, 22, 11, 23, 21, 25, 13, 29, 21, 34, 26, 37, 11, 40, 26, 44, 31, 45, 21, 54, 35, 54, 36, 55, 24, 65, 40, 59, 47, 70, 24, 71, 43, 72, 55, 81, 28, 74, 55, 88, 59, 90, 28, 93, 58, 91, 66, 96, 46, 110, 63, 100

Offset: 1

Torlach Rush, Dec 02 2020

nonn

a(n) <= A008805(n). This is because A008805(n) is the maximum number of points required to calculate a(n) and each point is located in the first quadrant.

			a(1) = 1 because there are 3 visible points from every point on the grid.
a(2) = 3 because 5 points are visible from every vertex of the grid, 7 points are visible from the midpoint of every edge of the grid, and 8 points are visible from the middle of the grid.
a(3) = 3 because 9 points are visible from every vertex of the grid, 11 points are visible from the inner points of every edge of the grid, and 12 points are visible from every inner point of the grid.

Eric Weisstein's World of Mathematics, Visible Point

Cf. A008805, A049687, A049691, A300778, A331771.

\\ n = side length, d = dimension
cdvps(n, d) ={my(m=Map());
  forvec(u=vector(d, i, [0, n\2]),
    my(c=0); forvec(v=[[t-n, t]|t<-u], c+=(gcd(v)==1));
    mapput(m, c, 1), 1);
  #m; }
a(n) = cdvps(n, 2)

A344533 Given a square forest of n X n trees, with rows and columns separated by 1 meter, a(n) is the number of trees visible to an observer halfway along one side of the forest, exactly one meter outside.

Original entry on oeis.org

1, 4, 7, 14, 17, 30, 33, 52, 51, 82, 81, 108, 105, 156, 143, 198, 183, 252, 231, 308, 267, 380, 339, 436, 383, 526, 461, 598, 525, 680, 595, 782, 663, 896, 767, 974, 839, 1118, 953, 1208, 1041, 1330, 1143, 1466, 1227, 1620, 1383, 1738, 1473, 1898, 1605, 2034

Offset: 1

John Mason, May 22 2021

nonn

This concept has been studied under the name "visible lattice points" although the usual version considers the points in an n X n grid that are visible from the origin. - Jeffrey Shallit, May 22 2021

			For example, if the forest contains 5 X 5 trees, the observer will see only 17, as 8 will be hidden.

John Mason, Table of n, a(n) for n = 1..10000

Cf. A049687, A049691, A124254, A300778.

cp's OEIS Frontend

A049687 Array T read by diagonals: T(i,j)=number of lines passing through (0,0) and at least one other lattice point (h,k) satisfying 0<=h<=i, 0<=k<=j.

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A339400 Mark each point on the n X n grid with the number of points that are visible from it; a(n) is the number of distinct values in the grid.

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Examples

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Crossrefs

Programs

PARI

A344533 Given a square forest of n X n trees, with rows and columns separated by 1 meter, a(n) is the number of trees visible to an observer halfway along one side of the forest, exactly one meter outside.

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