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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.

%I A096948 #41 Feb 26 2025 08:50:23
%S A096948 1,3,9,6,18,36,10,30,60,100,15,45,90,150,225,21,63,126,210,315,441,28,
%T A096948 84,168,280,420,588,784,36,108,216,360,540,756,1008,1296,45,135,270,
%U A096948 450,675,945,1260,1620,2025,55,165,330,550,825,1155,1540,1980,2475,3025
%N 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.
%C A096948 Table of products of triangular numbers A000217.
%C A096948 Because of symmetry it is sufficient to consider n X m rectangles with n >= m. A square is a special rectangle.
%H A096948 Paolo Xausa, <a href="/A096948/b096948.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of triangle, flattened).
%H A096948 Steve Chow, <a href="https://www.youtube.com/watch?v=Uq9OXC0Gzgw">Math for fun, how many rectangles?</a>, blackpenredpen on YouTube, Apr 14 2018.
%H A096948 Wolfdieter Lang, <a href="/A096948/a096948.txt">First 10 rows</a>.
%F A096948 T(n, m) = t(n)*t(m) if n>=m else 0, with the triangular numbers t(n):= A000217(n), n>=1.
%F A096948 G.f. for column m (without leading zeros): t(m)*(x/(1-x)^3 - Sum_{k=0..m-1} t(k)*x^k)/x^m, m>=1.
%e A096948 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.
%e A096948 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.
%e A096948 Triangle begins:
%e A096948    1,
%e A096948    3,   9,
%e A096948    6,  18,  36,
%e A096948   10,  30,  60, 100,
%e A096948   15,  45,  90, 150, 225,
%e A096948   21,  63, 126, 210, 315, 441,
%e A096948   28,  84, 168, 280, 420, 588, 784,
%e A096948   36, 108, 216, 360, 540, 756,1008,1296,
%e A096948   45, 135, 270, 450, 675, 945,1260,1620,2025,
%e A096948   55, 165, 330, 550, 825,1155,1540,1980,2475,3025,
%e A096948   (...)
%t A096948 Table[n*(n+1)*m*(m+1)/4, {n, 10}, {m, n}] (* _Paolo Xausa_, Feb 25 2025 *)
%o A096948 (PARI) T(n,m)=if(m>n,0,n*(n+1)*m*(m+1)/4) \\ _Charles R Greathouse IV_, Dec 14 2015
%Y A096948 Cf. A000217 (1st column), A045943 (2nd column), A028896 (3rd column), A124080 (4th col).
%K A096948 nonn,easy,tabl
%O A096948 1,2
%A A096948 _Wolfdieter Lang_, Jul 16 2004
%E A096948 Name edited by _M. F. Hasler_, Oct 22 2020