A114043 -id:A114043 - cp's OEIS frontend

A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2.

4, 8, 20, 12, 32, 52, 16, 48, 80, 124, 20, 64, 108, 168, 228, 24, 84, 144, 228, 312, 428, 28, 104, 180, 288, 396, 544, 692, 32, 128, 224, 360, 496, 684, 872, 1100, 36, 152, 268, 432, 596, 824, 1052, 1328, 1604, 40, 180, 320, 520, 720, 1000, 1280, 1620, 1960, 2396

Offset: 2

N. J. A. Sloane, Feb 12 2020

			Triangle begins:
4,
8, 20,
12, 32, 52,
16, 48, 80, 124,
20, 64, 108, 168, 228,
24, 84, 144, 228, 312, 428,
28, 104, 180, 288, 396, 544, 692,
32, 128, 224, 360, 496, 684, 872, 1100,
36, 152, 268, 432, 596, 824, 1052, 1328, 1604,
...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 12.
N. J. A. Sloane, Illustration for m=n=3

Cf. A332350, A332352, A331781, A332371, A332372, A332374.

For main diagonal see A332368.

# Maple code for sequences mentioned in Theorem 12 of Alekseyev et al. (2015).
VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
VS := proc(m,n) local a,i,j; a:=0; # A331781
for i from 1 to m-1 do for j from 1 to n-1 do
if gcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
c3 := (m,n) -> VR(m,n,2)+4; # A332367
for m from 2 to 12 do lprint([seq(c3(m,n),n=2..m)]); od:
[seq(c3(n,n)/4,n=2..40)]; # A332368
c4 := (m,n) -> VR(m,n,1)/2 - VR(m,n,2) - 3; # A332369
for m from 2 to 12 do lprint([seq(c4(m,n),n=2..m)]); od:
[seq(c4(n,n),n=2..40)]; # A332370
ct := (m,n) -> c3(m,n)+c4(m,n); # A332371
for m from 2 to 12 do lprint([seq(ct(m,n),n=2..m)]); od:
[seq(ct(n,n),n=2..40)]; # A114043
et := (m,n) -> VR(m,n,1) - VR(m,n,2)/2 - VS(m,n) - 2; # A332372
for m from 2 to 12 do lprint([seq(et(m,n),n=2..m)]); od:
[seq(et(n,n),n=2..40)]; # A332373
vt := (m,n) ->  et(m,n) - ct(m,n) +1; # A332374
for m from 2 to 12 do lprint([seq(vt(m,n),n=2..m)]); od:
[seq(vt(n,n),n=2..40)]; # A332375

A332371 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of cells in the partition for m >= n >= 2.

Original entry on oeis.org

7, 14, 29, 23, 50, 87, 34, 75, 132, 201, 47, 106, 189, 290, 419, 62, 141, 252, 387, 560, 749, 79, 182, 327, 504, 731, 980, 1283, 98, 227, 410, 633, 920, 1235, 1618, 2041, 119, 278, 503, 778, 1133, 1522, 1995, 2518, 3107, 142, 333, 604, 935, 1362, 1829, 2398, 3027, 3736, 4493

Offset: 2

N. J. A. Sloane, Feb 12 2020

Equals sum of triangles A332367 and A332369.

			Triangle begins:
7,
14, 29,
23, 50, 87,
34, 75, 132, 201,
47, 106, 189, 290, 419,
62, 141, 252, 387, 560, 749,
79, 182, 327, 504, 731, 980, 1283,
98, 227, 410, 633, 920, 1235, 1618, 2041,
119, 278, 503, 778, 1133, 1522, 1995, 2518, 3107,
...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 12.
N. J. A. Sloane, Illustration for m=n=3

Cf. A332350, A332352, A331781, A332367, A332372, A332374.

For main diagonal see A114043.

Maple
```
See A332367.
```

A115009 Array read by antidiagonals: let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)(n+1-j), then T(m,n) = 2mn+m+n+2V(m,n), for m >= 0, n >= 0.

Original entry on oeis.org

0, 1, 1, 2, 6, 2, 3, 13, 13, 3, 4, 22, 28, 22, 4, 5, 33, 49, 49, 33, 5, 6, 46, 74, 86, 74, 46, 6, 7, 61, 105, 131, 131, 105, 61, 7, 8, 78, 140, 188, 200, 188, 140, 78, 8, 9, 97, 181, 251, 289, 289, 251, 181, 97, 9, 10, 118, 226, 326, 386, 418, 386, 326, 226, 118, 10, 11, 141, 277

Offset: 0

N. J. A. Sloane, Feb 24 2006

This is the number of linear partitions of an m X n grid.

			The array begins:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
1, 6, 13, 22, 33, 46, 61, 78, 97, 118, ...
2, 13, 28, 49, 74, 105, 140, 181, 226, 277, ...
3, 22, 49, 86, 131, 188, 251, 326, 409, 502, ...
4, 33, 74, 131, 200, 289, 386, 503, 632, 777, ...
5, 46, 105, 188, 289, 418, 559, 730, 919, 1132, ...
6, 61, 140, 251, 386, 559, 748, 979, 1234, 1521, ...
7, 78, 181, 326, 503, 730, 979, 1282, 1617, 1994, ...
...

D. M. Acketa, J. D. Zunic: On the number of linear partitions of the (m,n)-grid. Inform. Process. Lett., 38 (3) (1991), 163-168. See Table A.1.
Jovisa Zunic, Note on the number of two-dimensional threshold functions, SIAM J. Discrete Math. Vol. 25 (2011), No. 3, pp. 1266-1268. See Equation (1.2).

Max A. Alekseyev. On the number of two-dimensional threshold functions. SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631. doi:10.1137/090750184

The second and third rows are A028872 and A358296.
The main diagonal is A141255 = A114043 - 1.
The lower triangle is A332351.
Cf. A114999, A114043, A115004, A115005, A115006, A115007, A115010, A115011.

V:=proc(m,n) local t1,i,j; t1:=0; for i from 1 to m do for j from 1 to n do if gcd(i,j)=1 then t1:=t1+(m+1-i)*(n+1-j); fi; od; od; t1; end; T:=(m,n)->(2*m*n+m+n+2*V(m,n));

V[m_, n_] := Sum[If[GCD[i, j] == 1, (m-i+1)*(n-j+1), 0], {i, 1, m}, {j, 1, n}]; T[m_, n_] := 2*m*n+m+n+2*V[m, n]; Table[T[m-n, n], {m, 0, 11}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

A177719 Number of line segments connecting exactly 3 points in an n X n grid of points.

Original entry on oeis.org

0, 0, 8, 24, 60, 112, 212, 344, 548, 800, 1196, 1672, 2284, 2992, 3988, 5128, 6556, 8160, 10180, 12424, 15068, 17968, 21604, 25576, 30092, 34976, 40900, 47288, 54500, 62224, 70972, 80296, 90740, 101824, 114700, 128344, 143212, 158896, 176836

Offset: 1

Seppo Mustonen, May 13 2010

nonn

a(n) is also the number of pairs of points visible to each other exactly through one point in an n X n grid of points.

Mathematica code below computes with j=1 also A114043(n)-1 and A141255(n) much more efficiently than codes/formulas currently presented for those sequences.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points [Local copy]

j=2;
a[n_]:=a[n]=If[n<=j,0,2*a1[n]-a[n-1]+R1[n]]
a1[n_]:=a1[n]=If[n<=j,0,2*a[n-1]-a1[n-1]+R2[n]]
R1[n_]:=R1[n]=If[n<=j,0,R1[n-1]+4*S[n]]
R2[n_]:=(n-1)*S[n]
S[n_]:=If[Mod[n-1,j]==0,EulerPhi[(n-1)/j],0]
Table[a[n],{n,1,50}]

{ A177719(n) = if(n<2, return(0)); 2*(n*(n-2) + sum(i=1,n-1,sum(j=1,n-1, (gcd(i,j)==2)*(n-i)*(n-j))) ); } \\ Max Alekseyev, Jul 08 2019

from sympy import totient
def A177719(n): return 4*((n-1)*(n-2) + sum(totient(i)*(n-2*i)*(n-i) for i in range(2,n//2+1))) # Chai Wah Wu, Aug 18 2021

a(n) = Sum_{-n < i,j < n; gcd(i,j)=2} (n-|i|)*(n-|j|)/2. For n>1, a(n) = 2 * ( n*(n-2) + Sum_{i,j=1..n-1; gcd(i,j)=2} (n-i)*(n-j) ). - Max Alekseyev, Jul 08 2019

a(n) = 4*((n-1)*(n-2) + Sum_{i=2..floor(n/2)} (n-2*i)*(n-i)*phi(i)). - Chai Wah Wu, Aug 18 2021

A331771 a(n) = Sum_{-n

Original entry on oeis.org

0, 12, 56, 172, 400, 836, 1496, 2564, 4080, 6212, 8984, 12788, 17488, 23644, 31112, 40148, 50912, 64172, 79448, 97868, 118912, 143108, 170504, 202500, 238080, 278700, 323864, 374508, 430272, 493380, 561832, 638692, 722656, 814604, 914360, 1023428

Offset: 1

N. J. A. Sloane, Feb 08 2020

nonn

a(n) = 8*A332612(n)+4*n*(n-1)+4*(n-1)^2. Also adding 2 to the terms of the present sequence gives (essentially) A114146. - N. J. A. Sloane, Mar 14 2020

Koplowitz, Jack, Michael Lindenbaum, and A. Bruckstein. "The number of digital straight lines on an N* N grid." IEEE Transactions on Information Theory 36.1 (1990): 192-197. (See I(n).)

Seiichi Manyama, Table of n, a(n) for n = 1..1000
M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See p. 158.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

When divided by 4 this becomes A115005, so this is a ninth sequence to add to the following list.
The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n.
Cf. A332612.

VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
[seq(VR(n,n,1),n=1..50)];

a[n_] := Sum[Boole[GCD[i, j] == 1] (n - Abs[i]) (n - Abs[j]), {i, -n + 1, n - 1}, {j, -n + 1, n - 1}];
Array[a, 36] (* Jean-François Alcover, Apr 19 2020 *)

from sympy import totient
def A331771(n): return 4*((n-1)*(2*n-1)+sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n))) # Chai Wah Wu, Aug 17 2021

a(n) = 4 * A115005(n).

a(n) = 4*((n-1)*(2n-1)+Sum_{i=2..n-1} (n-i)*(2*n-i)*phi(i)). - Chai Wah Wu, Aug 17 2021

A332351 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

Original entry on oeis.org

0, 1, 6, 2, 13, 28, 3, 22, 49, 86, 4, 33, 74, 131, 200, 5, 46, 105, 188, 289, 418, 6, 61, 140, 251, 386, 559, 748, 7, 78, 181, 326, 503, 730, 979, 1282, 8, 97, 226, 409, 632, 919, 1234, 1617, 2040, 9, 118, 277, 502, 777, 1132, 1521, 1994, 2517, 3106, 10, 141, 332, 603, 934, 1361, 1828, 2397, 3026, 3735, 4492

Offset: 1

N. J. A. Sloane, Feb 10 2020

This is the triangle in A332350, halved.

This triangle is the lower half of the array defined in A115009.

			Triangle begins:
0,
1, 6,
2, 13, 28,
3, 22, 49, 86,
4, 33, 74, 131, 200,
5, 46, 105, 188, 289, 418,
6, 61, 140, 251, 386, 559, 748,
7, 78, 181, 326, 503, 730, 979, 1282,
8, 97, 226, 409, 632, 919, 1234, 1617, 2040,
9, 118, 277, 502, 777, 1132, 1521, 1994, 2517, 3106,
...

Jovisa Zunic, Note on the number of two-dimensional threshold functions, SIAM J. Discrete Math. Vol. 25 (2011), No. 3, pp. 1266-1268. See Equation (1.2).

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)
M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. This sequence is f_1(m,n)/2.

The main diagonal is A141255, or A114043 - 1.
Cf. A115009, A332350, A332352.
This is the lower triangle of the array in A115009.

VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
for m from 1 to 12 do lprint(seq(VR(m,n,1)/2,n=1..m),); od:

A332351[m_,n_]:=Sum[If[CoprimeQ[i,j],2(m-i)(n-j),0],{i,m-1},{j,n-1}]+2m*n-m-n;Table[A332351[m,n],{m,15},{n,m}] (* Paolo Xausa, Oct 18 2023 *)

A114531 Number of intersections of a convex set with the n X n grid.

Original entry on oeis.org

1, 2, 16, 214, 2856, 33367, 349895, 3302046, 28597720, 229893484, 1730841042, 12296287989, 82945782429, 533994100001, 3295058878301, 19560004053448, 112049953456910, 621112646095769, 3339451666010949, 17451533356424837, 88807504851447189, 440798078343276894

Offset: 0

David Applegate, Feb 24 2006

nonn

This is a generalization of A114043, which can be viewed as the number of intersections of a halfspace with the n X n grid (divided by 2).

			a(3) = 214 (X's indicate points in intersection):
... X.. .X. ... XX. X.. X.. X.. XXX XX. XX. XX. X.. X.. .X. .X.
... ... ... .X. ... .X. ..X ... ... X.. .X. ..X .XX .X. .X. XX.
... ... ... ... ... ... ... .X. ... ... ... ... ... ..X .X. ...
-1- -4- -4- -1- -8- -4- -8- -8- -4- -4- -8- -8- -8- -2- -2- -4-
XXX XXX XX. XX. XX. XX. .X. X.. XXX XXX XX. XX. XX. XX. .X. XX.
X.. .X. XX. .X. .XX .X. XXX .XX XX. .X. .XX .XX XXX XX. XXX .X.
... ... ... .X. ... ..X ... .X. ... .X. ..X .X. ... ..X .X. .XX
-8- -4- -4- -8- -8- -8- -4- -4- -8- -4- -4- -8- -8- -4- -1- -4-
XXX XXX XXX XX. XX. XXX XXX XX. XXX XXX
XXX XX. XX. XXX XXX XXX XXX XXX XXX XXX
... X.. .X. .X. ..X X.. .X. .XX XX. XXX
-4- -4- -8- -4- -8- -8- -4- -2- -4- -1-

Peter J. Taylor, Python program to generate terms

Cf. A114043.

More terms from Peter J. Taylor, Feb 28 2019

A332612 a(n) = Sum_{ i=2..n-1, j=1..i-1, gcd(i,j)=1 } (n-i)*(n-j).

Original entry on oeis.org

0, 0, 2, 11, 32, 77, 148, 268, 442, 691, 1018, 1472, 2036, 2780, 3686, 4786, 6100, 7724, 9598, 11863, 14454, 17437, 20818, 24772, 29172, 34200, 39794, 46071, 52986, 60817, 69314, 78860, 89292, 100720, 113122, 126686, 141244, 157294, 174566, 193228, 213172, 234954, 258058, 283189, 309946, 338473, 368782, 401516, 436040

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 14 2020

nonn

Related to the number of linear dichotomies on a square grid.

A331771(n) = 8*a(n) + 4*n*(n-1) + 4*(n-1)^2.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Jack Koplowitz, Michael Lindenbaum, and A. Bruckstein, The number of digital straight lines on an NxN grid, IEEE Transactions on Information Theory 36.1 (1990): 192-197. (See I_1.)

The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n. The present sequence and A331771 could be added to this list.

I1 := proc(n) local a, i, j; a:=0;
for i from 2 to n-1 do for j from 1 to i-1 do
if igcd(i,j)=1 then a := a+(n-i)*(n-j); fi; od; od; a; end;
[seq(I1(n),n=1..40)];

a(n) = sum(i=2, n-1, sum(j=1, i-1, if (gcd(i,j)==1, (n-i)*(n-j)))); \\ Michel Marcus, Mar 14 2020

from sympy import totient
def A332612(n): return sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n))//2 # Chai Wah Wu, Aug 17 2021

a(n) = (Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i))/2. - Chai Wah Wu, Aug 17 2021

A355902 Start with a 2 X n array of squares, join every vertex on top edge to every vertex on bottom edge; a(n) = one-half the number of cells.

Original entry on oeis.org

0, 3, 10, 26, 56, 112, 196, 331, 522, 790, 1138, 1615, 2204, 2975, 3910, 5041, 6388, 8047, 9958, 12262, 14894, 17920, 21346, 25347, 29796, 34875, 40522, 46854, 53826, 61716, 70274, 79883, 90380, 101875, 114346, 127981, 142612, 158737, 176086, 194827, 214852, 236717, 259906, 285124, 311970, 340588, 370990, 403819, 438440, 475556

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Sep 05 2022

nonn

Note that this figure can be obtained by drawing an "equatorial" line through the middle of the strip of n adjacent rectangles in A306302. This cuts each of the 2n "equatorial" cells in A306302 in two. It follows that 2*a(n) = A306302(n) + 2*n, i.e. that a(n) = A306302(n)/2 + n. Note that there is an explicit formula for A306302(n) in terms of n. - Scott R. Shannon, Sep 06 2022.

This means the present sequence is one more member of the large class of sequences which are essentially the same as A115004 (see Cross-References). - N. J. A. Sloane, Sep 06 2022

Scott R. Shannon, Image for a(4) = 56. Note in this and other images the entire 2xn array is shown so the number of cells is twice a(n).
Scott R. Shannon, Image for a(6) = 196.
Scott R. Shannon, Image for a(10) = 1138.
Scott R. Shannon, Image for a(15) = 5041.
N. J. A. Sloane, Illustration for a(2) = 10.
N. J. A. Sloane, Illustration for a(3) = 26.

Cf. A356790, A306302, A355798, A290131, A331452.

The following nine sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n; A355902(n) = n + A306302(n)/2. - N. J. A. Sloane, Sep 06 2022

a(n) = A356790(2,n+2)/2 - 2.

A115010 Array read by antidiagonals: let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)(n+1-j), then T(m,n) = 2mn+m+n+2V(m,n), for m >= 1, n >= 1.

Original entry on oeis.org

6, 13, 13, 22, 28, 22, 33, 49, 49, 33, 46, 74, 86, 74, 46, 61, 105, 131, 131, 105, 61, 78, 140, 188, 200, 188, 140, 78, 97, 181, 251, 289, 289, 251, 181, 97, 118, 226, 326, 386, 418, 386, 326, 226, 118, 141, 277, 409, 503, 559, 559, 503, 409, 277, 141, 166, 332, 502, 632, 730

Offset: 1

N. J. A. Sloane, Feb 24 2006

Max A. Alekseyev, On the number of two-dimensional threshold functions, arXiv:math/0602511 [math.CO], 2006-2010; SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631. doi:10.1137/090750184

Cf. A114999, A114043, A115004, A115005, A115006, A115007.

V:=proc(m,n) local t1,i,j; t1:=0; for i from 1 to m do for j from 1 to n do if gcd(i,j)=1 then t1:=t1+(m+1-i)*(n+1-j); fi; od; od; t1; end; T:=(m,n)->(2*m*n+m+n+2*V(m,n));

V[m_, n_] := Sum[Boole[CoprimeQ[i, j]]*(m-i+1)*(n-j+1), {i, m}, {j, n}];
T[m_, n_] := 2*m*n + m + n + 2*V[m, n];
Table[T[m - n + 1, n], {m, 1, 12}, {n, 1, m}] // Flatten (* Jean-François Alcover, Nov 28 2017 *)

cp's OEIS Frontend

A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2.

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A332371 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of cells in the partition for m >= n >= 2.

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Maple

A115009 Array read by antidiagonals: let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)*(n+1-j), then T(m,n) = 2*m*n+m+n+2*V(m,n), for m >= 0, n >= 0.

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Maple

Mathematica

A177719 Number of line segments connecting exactly 3 points in an n X n grid of points.

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Mathematica

PARI

Python

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A331771 a(n) = Sum_{-n

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Mathematica

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A332351 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

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Maple

Mathematica

A114531 Number of intersections of a convex set with the n X n grid.

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A332612 a(n) = Sum_{ i=2..n-1, j=1..i-1, gcd(i,j)=1 } (n-i)*(n-j).

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A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2.

A332371 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of cells in the partition for m >= n >= 2.

A115009 Array read by antidiagonals: let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)(n+1-j), then T(m,n) = 2mn+m+n+2V(m,n), for m >= 0, n >= 0.

A115010 Array read by antidiagonals: let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)(n+1-j), then T(m,n) = 2mn+m+n+2V(m,n), for m >= 1, n >= 1.