A327141 a(n) is the number of different sizes of integer-sided rectangles which can be placed inside an n X n square.

1, 3, 6, 10, 16, 22, 29, 39, 48, 61, 72, 84, 101, 115, 135, 151, 174, 192, 211, 238, 259, 289, 312, 336, 370, 396, 433, 461, 501, 531, 562, 606, 639, 686, 721, 757, 808, 846, 900, 940, 981, 1039, 1082, 1143, 1188, 1252, 1299, 1347, 1415, 1465, 1536, 1588, 1641

Offset

1,2

Comments

Rectangles can be placed in any orientation and in particular their sides are not required to be parallel to the sides of the square.

A000217(n) enumerates the subset with sides not larger than n. For n < 5, a(n) = A000217(n).

Condition 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.

Links

Table of n, a(n) for n=1..53.

Formula

a(n) = A000217(n) + A327142(n).

Example

For n = 3 there are 6 distinct sizes of rectangle that can be placed into a 3 X 3 square:
...  ...  ...  ...  ...  ###
...  ...  ...  ##.  ###  ###
#.., ##., ###, ##., ###, ###
For n = 8 there are 36 distinct sizes of rectangle that can be placed with their sides parallel to the sides of the square; in addition, 9 X 1, 9 X 2 and 10 X 1 rectangles can also be placed. Hence a(8) = 39.

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;
for (int j=1; j<=i; j++)
for (int k=j; k<=i; k++)
{count++; }
//Diagonally placed 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;
}

Crossrefs

Cf. A000217, A327142.

Sequence in context: A310083 A310084 A025215 * A283503 A294481 A049697

Adjacent sequences: A327138 A327139 A327140 * A327142 A327143 A327144

Keyword

nonn

Author

Kirill Ustyantsev, Aug 23 2019

Status

approved