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 A289832 Triangle read by rows: T(n,k) = number of rectangles all of whose vertices lie on an (n+1) X (k+1) rectangular grid. 0
 1, 3, 10, 6, 20, 44, 10, 33, 74, 130, 15, 49, 110, 198, 313, 21, 68, 152, 276, 443, 640, 28, 90, 200, 364, 592, 866, 1192, 36, 115, 254, 462, 756, 1113, 1550, 2044, 45, 143, 314, 570, 935, 1385, 1944, 2586, 3305, 55, 174, 380, 688, 1129, 1680, 2370, 3172, 4081, 5078 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS T(n,k) is the number of rectangles (including squares) that can be drawn on an (n+1) X (k+1) grid. The diagonal of T(n,k) is the number of rectangles in a square lattice (A085582), i.e., T(n,n) = A085582(n+1). Column k=1 equals A000217. Column k=2 equals A140229 for n >= 3 as the only oblique rectangles are squares of side length sqrt(2), leading to the same formula. LINKS EXAMPLE Triangle T(n,k) begins: n/k  1    2    3    4     5     6     7     8     9    10 1    1 2    3   10 3    6   20   44 4   10   33   74  130 5   15   49  110  198   313 6   21   68  152  276   443   640 7   28   90  200  364   592   866  1192 8   36  115  254  462   756  1113  1550  2044 9   45  143  314  570   935  1385  1944  2586  3305 10  55  174  380  688  1129  1680  2370  3172  4081  5078 e.g., there are T(3,3) =  44 rectangles in a 4 X 4 lattice: There are A096948(3,3) = 36 rectangles whose sides are parallel to the axes; There are 4 squares with side length sqrt(2); There are 2 squares with side length sqrt(5); There are 2 rectangles with side lengths sqrt(2) X 2 sqrt(2). PROG (Python) from math import gcd def countObliques(a, b, c, d, n, k):     if(gcd(a, b) == 1): #avoid double counting         boundingBox={'width':(b * c) + (a * d), 'height':(a * c) + (b * d)}         if(boundingBox['width']

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Last modified June 29 18:25 EDT 2022. Contains 354913 sequences. (Running on oeis4.)