A096948 Triangular table read by rows: T(n,m) = number of rectangles found in an n X m rectangle built from 1 X 1 squares, 1 <= m <= n.

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

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

Table of n, a(n) for n=1..55.

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

Cf. A000217 (1st column), A045943 (2nd column), A028896 (3rd column), A124080 (4th col).

Sequence in context: A154593 A283439 A134693 * A224262 A223918 A224190

Adjacent sequences: A096945 A096946 A096947 * A096949 A096950 A096951

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Jul 16 2004

Extensions

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

Status

approved