A287392 -id:A287392 - cp's OEIS frontend

A360610 Triangle read by rows: T(n,k) is the number of squares of side length k that can be placed inside a square of side length n without overlap, 1 <= k <= n.

1, 4, 1, 9, 1, 1, 16, 4, 1, 1, 25, 4, 1, 1, 1, 36, 9, 4, 1, 1, 1, 49, 9, 4, 1, 1, 1, 1, 64, 16, 4, 4, 1, 1, 1, 1, 81, 16, 9, 4, 1, 1, 1, 1, 1, 100, 25, 9, 4, 4, 1, 1, 1, 1, 1, 121, 25, 9, 4, 4, 1, 1, 1, 1, 1, 1, 144, 36, 16, 9, 4, 4, 1, 1, 1, 1, 1, 1, 169, 36, 16, 9, 4, 4, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Torlach Rush, Feb 13 2023

T(n,k) is square 1 <= k <= n.
Alternative triangle construction: Write each column k as each square repreated k times.
T(*,1) is A000290.
T(*,2) is A008794.
T(*,3) is A211547.
T(*,4) is A295643(n+4).
T(*,5) is A287392(n+1).
Row sums of triangle are A222548.
This assumes the sides of the small squares are parallel to those of the large square.  If the small squares are allowed to be rotated, better packings may exist (see e.g. the Friedman link).

			Sum_{T(1,*)} = A222548(1) = 1;
Sum_{T(2,*)} = A222548(2) = 5;
Sum_{T(3,*)} = A222548(3) = 11.
Triangle begins:
    1;
    4,  1;
    9,  1, 1;
   16,  4, 1, 1;
   25,  4, 1, 1, 1;
   36,  9, 4, 1, 1, 1;
   49,  9, 4, 1, 1, 1, 1;
   64, 16, 4, 4, 1, 1, 1, 1;
   81, 16, 9, 4, 1, 1, 1, 1, 1;
  100, 25, 9, 4, 4, 1, 1, 1, 1, 1;
  ...

E. Friedman, Packing Unit Squares in Squares: A Survey and New Results, The Electronic Journal of Combinatorics 5 (2009), DS#7.

Cf. A000290, A008794, A211547, A222548, A287392, A295643.

Python
```
def T(n, k): return (n//k)**2
```

T(n,k) = floor(n/k)^2.

cp's OEIS Frontend

A360610 Triangle read by rows: T(n,k) is the number of squares of side length k that can be placed inside a square of side length n without overlap, 1 <= k <= n.

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