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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.

%I A360610 #51 Feb 19 2023 17:13:14
%S A360610 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,
%T A360610 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,
%U A360610 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
%N 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.
%C A360610 T(n,k) is square 1 <= k <= n.
%C A360610 Alternative triangle construction: Write each column k as each square repreated k times.
%C A360610 T(*,1) is A000290.
%C A360610 T(*,2) is A008794.
%C A360610 T(*,3) is A211547.
%C A360610 T(*,4) is A295643(n+4).
%C A360610 T(*,5) is A287392(n+1).
%C A360610 Row sums of triangle are A222548.
%C A360610 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).
%H A360610 E. Friedman, <a href="https://doi.org/10.37236/28">Packing Unit Squares in Squares: A Survey and New Results</a>, The Electronic Journal of Combinatorics 5 (2009), DS#7.
%F A360610 T(n,k) = floor(n/k)^2.
%e A360610 Sum_{T(1,*)} = A222548(1) = 1;
%e A360610 Sum_{T(2,*)} = A222548(2) = 5;
%e A360610 Sum_{T(3,*)} = A222548(3) = 11.
%e A360610 Triangle begins:
%e A360610     1;
%e A360610     4,  1;
%e A360610     9,  1, 1;
%e A360610    16,  4, 1, 1;
%e A360610    25,  4, 1, 1, 1;
%e A360610    36,  9, 4, 1, 1, 1;
%e A360610    49,  9, 4, 1, 1, 1, 1;
%e A360610    64, 16, 4, 4, 1, 1, 1, 1;
%e A360610    81, 16, 9, 4, 1, 1, 1, 1, 1;
%e A360610   100, 25, 9, 4, 4, 1, 1, 1, 1, 1;
%e A360610   ...
%o A360610 (Python)
%o A360610 def T(n, k): return (n//k)**2
%Y A360610 Cf. A000290, A008794, A211547, A222548, A287392, A295643.
%K A360610 nonn,tabl
%O A360610 1,2
%A A360610 _Torlach Rush_, Feb 13 2023