A331762 -id:A331762 - cp's OEIS frontend

A331761 a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=2} (n+1-i)*(n+1-j).

0, 1, 4, 15, 32, 71, 124, 211, 320, 499, 716, 999, 1328, 1799, 2340, 3023, 3792, 4767, 5852, 7135, 8544, 10319, 12260, 14471, 16864, 19775, 22916, 26467, 30272, 34587, 39188, 44347, 49824, 56195, 62948, 70311, 78080, 86975

Offset: 1

N. J. A. Sloane, Feb 04 2020

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms n=1..600 from N. J. A. Sloane)
M. A. Alekseyev. On the number of two-dimensional threshold functions. arXiv:math/0602511 [math.CO], 2006-2009; doi:10.1137/090750184, SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631.
M. A. Alekseyev, M. Basova and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions, SIAM J. Disc. Math. 29(1), 2015, pp. 157-165.

Cf. A115004.

The main diagonal of A331762.

Table[Sum[Boole[GCD[i, j] == 2] (n + 1 - i) (n + 1 - j), {i, n}, {j, n}], {n, 38}] (* Michael De Vlieger, Feb 04 2020 *)

from sympy import totient
def A331761(n): return (n-1)**2 + 2*sum(totient(i)*(n+1-2*i)*(n+1-i) for i in range(2,n//2+1)) # Chai Wah Wu, Aug 16 2021

Conjecture: As n -> oo, a(n) ~ C*n^4/Pi^2, where C is about 0.3775. - N. J. A. Sloane, Jul 03 2020

a(n) = (n-1)^2 + 2*Sum_{i=2..floor(n/2)} (n+1-2*i)*(n+1-i)*phi(i). - Chai Wah Wu, Aug 16 2021

A335679 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of edges in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

Original entry on oeis.org

1, 3, 3, 5, 8, 5, 7, 15, 15, 7, 9, 24, 28, 24, 9, 11, 35, 47, 47, 35, 11, 13, 48, 69, 80, 69, 48, 13, 15, 63, 97, 119, 119, 97, 63, 15, 17, 80, 128, 170, 178, 170, 128, 80, 17, 19, 99, 165, 225, 257, 257, 225, 165, 99, 19, 21, 120, 205, 292, 340, 372, 340, 292, 205, 120, 21

Offset: 1

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

The case m=n (the main diagonal) is dealt with in A331757. A306302 has illustrations for the diagonal case for m = 1 to 15.

Also A335678 has colored illustrations for many values of m and n.

			The initial rows of the array are:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ...
3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, ...
5, 15, 28, 47, 69, 97, 128, 165, 205, 251, 300, 355, ...
7, 24, 47, 80, 119, 170, 225, 292, 365, 448, 537, 638, ...
9, 35, 69, 119, 178, 257, 340, 443, 555, 683, 819, 975, ...
11, 48, 97, 170, 257, 372, 493, 644, 809, 998, 1197, 1426, ...
13, 63, 128, 225, 340, 493, 654, 857, 1078, 1331, 1595, 1901, ...
15, 80, 165, 292, 443, 644, 857, 1124, 1415, 1748, 2095, 2498, ...
17, 99, 205, 365, 555, 809, 1078, 1415, 1782, 2203, 2640, 3149, ...
19, 120, 251, 448, 683, 998, 1331, 1748, 2203, 2724, 3265, 3896, ...
21, 143, 300, 537, 819, 1197, 1595, 2095, 2640, 3265, 3914, 4673, ...
...
The initial antidiagonals are:
1
3, 3
5, 8, 5
7, 15, 15, 7
9, 24, 28, 24, 9
11, 35, 47, 47, 35, 11
13, 48, 69, 80, 69, 48, 13
15, 63, 97, 119, 119, 97, 63, 15
17, 80, 128, 170, 178, 170, 128, 80, 17
19, 99, 165, 225, 257, 257, 225, 165, 99, 19
21, 120, 205, 292, 340, 372, 340, 292, 205, 120, 21
23, 143, 251, 365, 443, 493, 493, 443, 365, 251, 143, 23
...

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.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
Index entries for sequences related to stained glass windows

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
For the diagonal case see A306302 and A331757.
Cf. A114999, A331762.

Comment from Max Alekseyev, Jun 28 2020 (Start):
T(m,n) = 2*A114999(m-1,n-1) - A331762(m-1,n-1) + m*n + m + n - 2 for all m, n >= 1.  This follows from the Alekseyev-Basova-Zolotykh (2015) article.
Proof: Here is the appropriate modification of the corresponding comment in A306302: Assuming that K(m,n) has vertices at (i,0) and (j,1), for i=0..m-1 and j=0..n-1, the projective map (x,y) -> ((1-y)/(x+1), y/(x+1)) maps K(m,n) to the partition of the right triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. (End)

A335680 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

Original entry on oeis.org

2, 3, 3, 4, 5, 4, 5, 8, 8, 5, 6, 12, 13, 12, 6, 7, 17, 21, 21, 17, 7, 8, 23, 30, 35, 30, 23, 8, 9, 30, 42, 51, 51, 42, 30, 9, 10, 38, 55, 73, 75, 73, 55, 38, 10, 11, 47, 71, 96, 109, 109, 96, 71, 47, 11, 12, 57, 88, 125, 143, 159, 143, 125, 88, 57, 12, 13, 68, 108, 156, 187, 209, 209, 187, 156, 108, 68, 13

Offset: 1

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

The case m=n (the main diagonal) is dealt with in A331755. A306302 has illustrations for the diagonal case for m = 1 to 15.

Also A335678 has colored illustrations for many values of m and n.

			The initial rows of the array are:
  2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
  3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, ...
  4, 8, 13, 21, 30, 42, 55, 71, 88, 108, 129, 153, ...
  5, 12, 21, 35, 51, 73, 96, 125, 156, 192, 230, 274, ...
  6, 17, 30, 51, 75, 109, 143, 187, 234, 289, 346, 413, ...
  7, 23, 42, 73, 109, 159, 209, 274, 344, 426, 510, 609, ...
  8, 30, 55, 96, 143, 209, 275, 362, 455, 564, 674, 805, ...
  9, 38, 71, 125, 187, 274, 362, 477, 600, 744, 889, 1062, ...
  10, 47, 88, 156, 234, 344, 455, 600, 755, 937, 1119, 1337, ...
  11, 57, 108, 192, 289, 426, 564, 744, 937, 1163, 1389, 1660, ...
  12, 68, 129, 230, 346, 510, 674, 889, 1119, 1389, 1659, 1984, ...
  ...
The initial antidiagonals are:
  2
  3, 3
  4, 5, 4
  5, 8, 8, 5
  6, 12, 13, 12, 6
  7, 17, 21, 21, 17, 7
  8, 23, 30, 35, 30, 23, 8
  9, 30, 42, 51, 51, 42, 30, 9
  10, 38, 55, 73, 75, 73, 55, 38, 10
  11, 47, 71, 96, 109, 109, 96, 71, 47, 11
  12, 57, 88, 125, 143, 159, 143, 125, 88, 57, 12
  ...

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.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
Index entries for sequences related to stained glass windows

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
For the diagonal case see A306302 and A331755.
Cf. A114999, A331762.

Comment from Max Alekseyev, Jun 28 2020 (Start):
T(m,n) = A114999(m-1,n-1) - A331762(m-1,n-1) + m + n for all m, n >= 1. This follows from the Alekseyev-Basova-Zolotykh (2015) article.
Proof: Here is the appropriate modification of the corresponding comment in A306302: Assuming that K(m,n) has vertices at (i,0) and (j,1), for i=0..m-1 and j=0..n-1, the projective map (x,y) -> ((1-y)/(x+1), y/(x+1)) maps K(m,n) to the partition of the right triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. (End)
Max Alekseyev's formula is an analog of Proposition 9 of Legendre (2009), and gives an explicit formula for this array. - N. J. A. Sloane, Jun 30 2020

A335683 Array read by antidiagonals: T(n,k) (n >= 1, k >= 1) = Sum_{i=1..n, j=1..k, gcd(i,j)=2} (n+1-i)*(k+1-j).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 4, 4, 4, 0, 0, 6, 8, 8, 6, 0, 0, 9, 12, 15, 12, 9, 0, 0, 12, 18, 22, 22, 18, 12, 0, 0, 16, 24, 33, 32, 33, 24, 16, 0, 0, 20, 32, 44, 48, 48, 44, 32, 20, 0, 0, 25, 40, 58, 64, 71, 64, 58, 40, 25, 0, 0, 30, 50, 72, 84, 94, 94, 84, 72, 50, 30, 0

Offset: 1

N. J. A. Sloane, Jul 02 2020

			The array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, ...
0, 2, 4, 8, 12, 18, 24, 32, 40, 50, 60, 72, ...
0, 4, 8, 15, 22, 33, 44, 58, 72, 90, 108, 129, ...
0, 6, 12, 22, 32, 48, 64, 84, 104, 130, 156, 186, ...
0, 9, 18, 33, 48, 71, 94, 123, 152, 190, 228, 271, ...
0, 12, 24, 44, 64, 94, 124, 162, 200, 250, 300, 356, ...
0, 16, 32, 58, 84, 123, 162, 211, 260, 325, 390, 462, ...
0, 20, 40, 72, 104, 152, 200, 260, 320, 400, 480, 568, ...
0, 25, 50, 90, 130, 190, 250, 325, 400, 499, 598, 707, ...
0, 30, 60, 108, 156, 228, 300, 390, 480, 598, 716, 846, ...
0, 36, 72, 129, 186, 271, 356, 462, 568, 707, 846, 999, ...
...
The initial antidiagonals are:
[0]
[0, 0]
[0, 1, 0]
[0, 2, 2, 0]
[0, 4, 4, 4, 0]
[0, 6, 8, 8, 6, 0]
[0, 9, 12, 15, 12, 9, 0]
[0, 12, 18, 22, 22, 18, 12, 0]
[0, 16, 24, 33, 32, 33, 24, 16, 0]
[0, 20, 32, 44, 48, 48, 44, 32, 20, 0]
[0, 25, 40, 58, 64, 71, 64, 58, 40, 25, 0]
[0, 30, 50, 72, 84, 94, 94, 84, 72, 50, 30, 0]
...

A331762 is the same array displayed as a triangle.

cp's OEIS Frontend

A331761 a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=2} (n+1-i)*(n+1-j).

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A335679 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of edges in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

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Formula

A335680 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A335683 Array read by antidiagonals: T(n,k) (n >= 1, k >= 1) = Sum_{i=1..n, j=1..k, gcd(i,j)=2} (n+1-i)*(k+1-j).

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Author

Keywords

Examples

Crossrefs