A327142 a(n) is the number of different sizes of integer-sided rectangles which can be placed inside an n X n square and with length greater than n.

0, 0, 0, 0, 1, 1, 1, 3, 3, 6, 6, 6, 10, 10, 15, 15, 21, 21, 21, 28, 28, 36, 36, 36, 45, 45, 55, 55, 66, 66, 66, 78, 78, 91, 91, 91, 105, 105, 120, 120, 120, 136, 136, 153, 153, 171, 171, 171, 190, 190, 210, 210, 210, 231, 231, 253, 253, 276, 276, 276, 300, 300, 325, 325

Offset

1,8

Comments

Conditions for rectangles L x W which have length L > n: n - L/sqrt(2) > W/sqrt(2) where L/sqrt(2) and W/sqrt(2) are projections on the n X n square's sides.

If a rectangle with sides k X m fits in an n X n square then it fits by putting a parallel (without loss of generality, the longest) on the diagonal of the n X n square. Only the sum k + m really matters in order to see if the k X m rectangle fits in the square; it fits if k + m < sqrt(2) * n. - David A. Corneth, Aug 24 2019

Links

Table of n, a(n) for n=1..64.

Formula

a(n) = binomial(floor(n * sqrt(2)) - n, 2). - David A. Corneth, Aug 24 2019

a(n) = A327141(n) - A000217(n).

Example

For n = 1, 2, 3, 4 we cannot place rectangles with side length L > n.
For n = 5 we can place a 6 X 1 rectangle inside a 5 X 5 square, so a(5) = 1.
For n = 8 we can place 9 X 1, 9 X 2 and 10 X 1 rectangles inside an 8 X 8 square, so a(8) = 3.

Prog

(C++)
#include <iostream>
#include <cmath>
#include <vector>
using namespace std;
int main() {
int n;
cin>>n;
vector <int> v;
for (int i=1; i<=n; i++)
{int count=0;
// ii-length, jj-width of diagonal rectangle
for (int ii=i+1; ii<=int(sqrt(2)*i); ii++)
     for (int jj=1; jj<=i; jj++)
          if ((double(i)-double(ii)/sqrt(2))-double(jj)/sqrt(2)>0)
                    {count++; }
v.push_back(count);
}
for (int i=0; i<v.size(); i++)
cout << v[i] << ", ";
return 0;
}
(PARI) a(n) = {binomial(floor(n * sqrt(2)) - n, 2)} \\ David A. Corneth, Aug 24 2019

Crossrefs

Cf. A080754, A327141.

Sequence in context: A160745 A105676 A127739 * A175394 A070318 A257537

Adjacent sequences: A327139 A327140 A327141 * A327143 A327144 A327145

Keyword

nonn,easy

Author

Kirill Ustyantsev, Aug 23 2019

Status

approved