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A096948
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Table read by antidiagonals: T(n,m) = number of rectangles found in an n X m rectangle built from 1 X 1 squares.
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2
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Table of products of triagonal numbers A000217.
Because of symmetry it is sufficient to consider n X m rectangles with n>=m. A square is a special rectangle.
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LINKS
| W. Lang, First 10 rows.
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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.
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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.
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CROSSREFS
| Sequence in context: A131954 A154593 A134693 * A179483 A016676 A001148
Adjacent sequences: A096945 A096946 A096947 * A096949 A096950 A096951
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Jul 16 2004
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