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

1,2

Comments

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).

Links

Table of n, a(n) for n=1..91.

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

Formula

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

Example

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;
  ...

Prog

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

Crossrefs

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

Sequence in context: A176215 A364016 A143469 * A331147 A208508 A123726

Adjacent sequences: A360607 A360608 A360609 * A360611 A360612 A360613

Keyword

nonn,tabl

Author

Torlach Rush, Feb 13 2023

Status

approved