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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. 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

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

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

EXAMPLE

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.

--------------------

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-Fran├žois Alcover, Apr 25 2016, after Max Alekseyev *)

CROSSREFS

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

Cf. A099957.

Sequence in context: A079796 A242727 A229795 * A166189 A001779 A257201

Adjacent sequences:  A114040 A114041 A114042 * A114044 A114045 A114046

KEYWORD

nonn,nice

AUTHOR

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

EXTENSIONS

More terms from Max Alekseyev, Feb 22 2006

STATUS

approved

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Last modified June 28 20:39 EDT 2017. Contains 288840 sequences.