OFFSET

1,2

COMMENTS

Table of products of triangular numbers A000217.

Because of symmetry it is sufficient to consider n X m rectangles with n >= m. A square is a special rectangle.

LINKS

Steve Chow, Math for fun, how many rectangles?, blackpenredpen on YouTube, Apr 14 2018.

Wolfdieter Lang, First 10 rows.

FORMULA

T(n, m) = t(n)*t(m) if n>=m else 0, with the triangular numbers t(n):= A000217(n), n>=1.

G.f. for column m (without leading zeros): t(m)*(x/(1-x)^3 - sum(t(k)*x^k, k=0..m-1)/x^m, m>=1.

EXAMPLE

T(2,2) = 9 because in a 2 X 2 square there are four 1 X 1 squares, two 1 X 2 rectangles, two 2 X 1 rectangles and one 2 X 2 square: 4 + 2 + 2 + 1 =9.

T(3,2) = 18 = t(3)*t(2) because in a 3 X 2 rectangle there are six 1 X 1 squares, three 1 X 2 rectangles, four 2 X 1 rectangles, two 3 X 1 rectangles, two 2 X 2 squares and one 3 X 2 rectangle: 6 + 3 + 4 + 2 + 2 + 1 = 9 + 9 = 18.

1,

3, 9,

6, 18, 36,

10, 30, 60, 100,

15, 45, 90, 150, 225,

21, 63, 126, 210, 315, 441,

28, 84, 168, 280, 420, 588, 784,

36, 108, 216, 360, 540, 756,1008,1296,

45, 135, 270, 450, 675, 945,1260,1620,2025,

55, 165, 330, 550, 825,1155,1540,1980,2475,3025,

(...)

PROG

(PARI) T(n, m)=if(m>n, 0, n*(n+1)*m*(m+1)/4) \\ Charles R Greathouse IV, Dec 14 2015

CROSSREFS

KEYWORD

AUTHOR

Wolfdieter Lang, Jul 16 2004

EXTENSIONS

Name edited by M. F. Hasler, Oct 22 2020

STATUS

approved