A114043 Take an n X n square grid of points in the plane; a(n) = number of ways to divide the points into two sets using a straight line.

1, 7, 29, 87, 201, 419, 749, 1283, 2041, 3107, 4493, 6395, 8745, 11823, 15557, 20075, 25457, 32087, 39725, 48935, 59457, 71555, 85253, 101251, 119041, 139351, 161933, 187255, 215137, 246691, 280917, 319347, 361329, 407303

Offset: 1

Author

Ugo Merlone (merlone(AT)econ.unito.it) and N. J. A. Sloane, Feb 22 2006

Keywords

• nonn
• nice

Comments

Also, half of the number of two-dimensional threshold functions (A114146).

The line may not pass through any point. This is the "labeled" version - rotations and reflections are not taken into account (cf. A116696).

The number of ways to divide a (2n) X (2n) grid into two sets of equal size is given by 2*A099957(n). - David Applegate, Feb 23 2006

All terms are odd: the line that misses the grid contributes 1 to the total and all other lines contribute 2, 4 or 8, so the total must be odd.

What can be said about the 3-D generalization? - Max Alekseyev, Feb 27 2006

Examples

			Examples: the two sets are indicated by X's and o's.
a(2) = 7:
XX oX Xo XX XX oo oX
XX XX XX Xo oX XX oX
--------------------
a(3) = 29:
XXX oXX ooX ooo ooX ooo
XXX XXX XXX XXX oXX oXX
XXX XXX XXX XXX XXX XXX
-1- -4- -8- -4- -4- -8- Total = 29
--------------------
a(4)= 87:
XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX
XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX
XXXX XXXX XXXX XXXX XXXX XXXo XXXo XXXo XXoo XXoo
XXXX XXXo XXoo Xooo oooo XXoo Xooo oooo Xooo oooo
--1- --4- --8- --8- --4- --4- --8- --8- --8- --8-
XXXX XXXX XXXX XXXX XXXX
XXXo XXXX XXXX XXXo XXXo
XXoo Xooo oooo Xooo XXoo
Xooo oooo oooo oooo oooo
--4- --8- --2- --4- --8- Total = 87.
--------------------
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Max A. Alekseyev. On the number of two-dimensional threshold functions, arXiv:math/0602511 [math.CO], 2006-2010; 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 Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Eq. (11).
• N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Crossrefs

Cf. A114499, A115004, A115005, A116696 (unlabeled case), A114531, A114146.

Cf. A099957.

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. - N. J. A. Sloane, Feb 04 2020

Programs

Mathematica

a[n_] := 2*Sum[(n - i)*(n - j)*Boole[CoprimeQ[i, j]], {i, 1, n - 1}, {j, 1, n - 1}] + 2*n^2 - 2*n + 1; Array[a, 40] (* Jean-François Alcover, Apr 25 2016, after Max Alekseyev *)

Python

from sympy import totient
def A114043(n): return 4*n**2-6*n+3 + 2*sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n)) # Chai Wah Wu, Aug 15 2021

Formula

Let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)*(n+1-j); then a(n+1) = 2*(n^2 + n + V(n,n)) + 1. - Max Alekseyev, Feb 22 2006

a(n) ~ (3/Pi^2) * n^4. - Max Alekseyev, Feb 22 2006

a(n) = A141255(n) + 1. - T. D. Noe, Jun 17 2008

a(n) = 4*n^2 - 6*n + 3 + 2*Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i). - Chai Wah Wu, Aug 15 2021

Extensions

More terms from Max Alekseyev, Feb 22 2006