A107348 -id:A107348 - cp's OEIS frontend

A295707 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of lines through at least 2 points of an n X k grid of points.

0, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 18, 20, 18, 1, 1, 27, 35, 35, 27, 1, 1, 38, 52, 62, 52, 38, 1, 1, 51, 75, 93, 93, 75, 51, 1, 1, 66, 100, 136, 140, 136, 100, 66, 1, 1, 83, 131, 181, 207, 207, 181, 131, 83, 1, 1, 102, 164, 238, 274, 306, 274, 238, 164, 102, 1

Offset: 1

Seiichi Manyama, Nov 26 2017

			Square array begins:
   0,  1,  1,   1,   1, ...
   1,  6, 11,  18,  27, ...
   1, 11, 20,  35,  52, ...
   1, 18, 35,  62,  93, ...
   1, 27, 52,  93, 140, ...
   1, 38, 75, 136, 207, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points [Local copy]

Columns k=2..10 give A160842, A160843, A160844, A160845, A160846, A160847, A160848, A160849, A160850.

Main diagonal gives A018808. Reading up to the diagonal gives A107348.

A[n_, k_] := (1/2)(f[n, k, 1] - f[n, k, 2]);
f[n_, k_, m_] := Sum[If[GCD[mx/m, my/m] == 1, (n - Abs[mx])(k - Abs[my]), 0], {mx, -n, n}, {my, -k, k}];
Table[A[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 04 2023 *)

A(n,k) = (1/2) * (f(n,k,1) - f(n,k,2)), where f(n,k,m) = Sum ((n-|m*x|)*(k-|m*y|)); -n < m*x < n, -k < m*y < k, (x,y)=1.

A160842 Number of lines through at least 2 points of a 2 X n grid of points.

Original entry on oeis.org

0, 1, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502

Offset: 0

Seppo Mustonen, May 28 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
S. Mustonen, On lines and their intersection points in a rectangular grid of points
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points [Local copy]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A295707, A107348.

[0,1] cat [n^2 + 2: n in [2..100]]; // G. C. Greubel, Apr 30 2018

a[n_]:=If[n<2,n,n^2+2] Table[a[n],{n,0,50}]
Join[{0,1},Range[2,50]^2+2] (* Harvey P. Dale, Feb 06 2015 *)

Vec(-x*(2*x^3-4*x^2+3*x+1) / (x-1)^3 + O(x^100)) \\ Colin Barker, May 24 2015

a(n) = n^2 + 2 = A059100(n) = A010000(n) for n > 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. - Colin Barker, May 24 2015
G.f.: -x*(2*x^3 - 4*x^2 + 3*x + 1) / (x-1)^3. - Colin Barker, May 24 2015
Sum_{n>=1} 1/a(n) = Pi * coth(sqrt(2)*Pi) / 2^(3/2) - 1/4. - Vaclav Kotesovec, May 01 2018

A160843 Number of lines through at least 2 points of a 3 X n grid of points.

Original entry on oeis.org

0, 1, 11, 20, 35, 52, 75, 100, 131, 164, 203, 244, 291, 340, 395, 452, 515, 580, 651, 724, 803, 884, 971, 1060, 1155, 1252, 1355, 1460, 1571, 1684, 1803, 1924, 2051, 2180, 2315, 2452, 2595, 2740, 2891, 3044, 3203, 3364, 3531, 3700, 3875, 4052, 4235, 4420

Offset: 0

Seppo Mustonen, May 28 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
S. Mustonen, On lines and their intersection points in a rectangular grid of points
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

3rd row/column of A107348, A295707.

[0, 1] cat [2*n^2 + 3 - n mod 2: n in [2..100]]; // G. C. Greubel, Apr 30 2018

a[n_]:=If[n<2,n,2*n^2+3-Mod[n,2]] Table[a[n],{n,0,47}]
Join[{0, 1}, LinearRecurrence[{2, 0, -2, 1}, {11, 20, 35, 52}, 20]] (* G. C. Greubel, Apr 30 2018 *)

Vec(-x*(3*x^4-3*x^3-2*x^2+9*x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, May 24 2015

a(n) = 2*n^2 + 3 - n mod 2.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 5. - Colin Barker, May 24 2015
G.f.: -x*(3*x^4 - 3*x^3 - 2*x^2 + 9*x + 1) / ((x-1)^3*(x+1)). - Colin Barker, May 24 2015

A160844 Number of lines through at least 2 points of a 4 X n grid of points.

Original entry on oeis.org

0, 1, 18, 35, 62, 93, 136, 181, 238, 299, 370, 445, 532, 621, 722, 827, 942, 1061, 1192, 1325, 1470, 1619, 1778, 1941, 2116, 2293, 2482, 2675, 2878, 3085, 3304, 3525, 3758, 3995, 4242, 4493, 4756, 5021, 5298, 5579, 5870, 6165, 6472, 6781, 7102, 7427, 7762

Offset: 0

Seppo Mustonen, May 28 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
S. Mustonen, On lines and their intersection points in a rectangular grid of points
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

4th row/column of A107348, A295707.

I:=[18, 35, 62, 93, 136, 181]; [0,1] cat [n le 6 select I[n] else Self(n-1) +Self(n-2) -Self(n-4) -Self(n-5) +Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 30 2018

a[0]=0; a[1]=1; a[2]=18; a[3]=35; a[n_]:=a[n]=a[n]=2*a[n-1]-a[n-2]+R[n] c4={10,4,12,2,12,4}; R[n_]:=c4[[Mod[n+2,6]+1]] Table[a[n],{n,0,46}]
Join[{0,1}, LinearRecurrence[{1,1,0,-1,-1,1}, {18, 35, 62, 93, 136, 181}, 50]] (* G. C. Greubel, Apr 30 2018 *)

my(x='x+O('x^30)); concat([0], Vec(x*(4*x^6-3*x^4+9*x^3+16*x^2+ 17*x+1 )/((1-x)^3*(x+1)*(x^2+x+1)))) \\ G. C. Greubel, Apr 30 2018

a(n) = 2*a(n-1) - a(n-2) + C(mod(n+2,6) + 1), C=(10,4,12,2,12,4), for n >= 4.
From Colin Barker, May 24 2015: (Start)
  a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 5.
  G.f.: x*(4*x^6 - 3*x^4 + 9*x^3 + 16*x^2 + 17*x + 1) / ((1-x)^3*(x + 1)*(x^2 + x + 1)).
(End)

A160845 Number of lines through at least 2 points of a 5 X n grid of points.

Original entry on oeis.org

0, 1, 27, 52, 93, 140, 207, 274, 361, 454, 563, 676, 809, 944, 1099, 1258, 1433, 1614, 1815, 2016, 2237, 2464, 2707, 2954, 3221, 3490, 3779, 4072, 4381, 4696, 5031, 5366, 5721, 6082, 6459, 6840, 7241, 7644, 8067, 8494, 8937, 9386, 9855, 10324, 10813

Offset: 0

Seppo Mustonen, May 28 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
S. Mustonen, On lines and their intersection points in a rectangular grid of points

5th row/column of A107348, A295707.

m=5;
a[0]=0; a[1]=1;
a[2]=m^2+2;
a[3]=2*m^2+3-Mod[m,2];
a[n_]:=a[n]=2*a[n-1]-a[n-2]+2*p1[m,n]+2*p4[m,n]
p1[m_,n_]:=Sum[p2[m,n,y], {y,1,m-1}]
p2[m_,n_,y_]:=If[GCD[y,n-1]==1,m-y,0]
p[i_]:=If[i>0,i,0]
p2[m_,n_,x_,y_]:=p2[m,n,x,y]=(n-x)*(m-y)-p[n-2*x]*p[m-2*y]
p3[m_,n_,x_,y_]:=p2[m,n,x,y]-2*p2[m,n-1,x,y]+p2[m,n-2,x,y]
p4[m_,n_]:=p4[m,n]=If[Mod[n,2]==0,0,p42[m,n]]
p42[m_,n_]:=p42[m,n]=Sum[p43[m,n,y], {y,1,m-1}]
p43[m_,n_,y_]:=If[GCD[(n-1)/2,y]==1,p3[m,n,(n-1)/2,y],0]
Table[a[n],{n,0,44}]

a(n) = (1/2)*(f(m,n,1)-f(m,n,2)) where f(m,n,k) = Sum((n-|kx|)*(m-|ky|)); -n < kx < n, -m < ky < m, (x,y)=1, m=5.
For another more efficient formula, see Mathematica code below.
Conjectures from Colin Barker, May 24 2015: (Start)
a(n) = a(n-1) + a(n-3) - a(n-5) - a(n-7) + a(n-8) for n > 7.
G.f.: x*(5*x^8 + x^6 + 16*x^5 + 20*x^4 + 40*x^3 + 25*x^2 + 26*x + 1) / ((1 - x)^3*(x + 1)*(x^2 + 1)*(x^2 + x + 1)).
(End)

cp's OEIS Frontend

A295707 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of lines through at least 2 points of an n X k grid of points.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A160842 Number of lines through at least 2 points of a 2 X n grid of points.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A160843 Number of lines through at least 2 points of a 3 X n grid of points.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A160844 Number of lines through at least 2 points of a 4 X n grid of points.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A160845 Number of lines through at least 2 points of a 5 X n grid of points.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula