

A130684


Triangle read by rows: T(n,k) = number of squares (not necessarily orthogonal) all of whose vertices lie in an (n + 1) X (k + 1) square lattice.


2



1, 2, 6, 3, 10, 20, 4, 14, 30, 50, 5, 18, 40, 70, 105, 6, 22, 50, 90, 140, 196, 7, 26, 60, 110, 175, 252, 336, 8, 30, 70, 130, 210, 308, 420, 540, 9, 34, 80, 150, 245, 364, 504, 660, 825, 10, 38, 90, 170, 280, 420, 588, 780, 990, 1210, 11, 42, 100, 190, 315, 476, 672
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OFFSET

1,2


COMMENTS

Reading down the diagonal gives A002415.


LINKS



FORMULA

T(n, k) = k*(k+1)*(k+2)*(2*n  k + 1)/12 (k <= n).


EXAMPLE

T(2, 2) = 6 because there are 6 squares all of whose vertices lie in a 3 X 3 lattice: four squares of side length 1, one square of side length 2 and one nonorthogonal square of side length the square root of 2.
Triangle begins:
1;
2, 6;
3, 10, 20;
4, 14, 30, 50;
5, 18, 40, 70, 105;
6, 22, 50, 90, 140, 196;
7, 26, 60, 110, 175, 252, 336;
...


PROG



CROSSREFS

Cf. A002415. For squares whose edges are required to be parallel to the edges of the large square, see A082652.


KEYWORD



AUTHOR



STATUS

approved



