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A082652 Triangle read by rows: T(n,k) is the number of squares that can be found in a k X n rectangular grid of little squares, for 1 <= k <= n.

%I A082652 #46 Jan 03 2025 09:37:26
%S A082652 1,2,5,3,8,14,4,11,20,30,5,14,26,40,55,6,17,32,50,70,91,7,20,38,60,85,
%T A082652 112,140,8,23,44,70,100,133,168,204,9,26,50,80,115,154,196,240,285,10,
%U A082652 29,56,90,130,175,224,276,330,385,11,32,62,100,145,196,252,312,375,440,506
%N A082652 Triangle read by rows: T(n,k) is the number of squares that can be found in a k X n rectangular grid of little squares, for 1 <= k <= n.
%C A082652 Here the squares being counted have sides parallel to the gridlines; for all squares, see A130684.
%C A082652 T(n,k) also is the total number of balls in a pyramid of balls on an n X k rectangular base. - _N. J. A. Sloane_, Nov 17 2007. For example, if the base is 4 X 2, the total number of balls is 4*2 + 3*1 = 11 = T(4,2).
%C A082652 Row sums give A001296. - _Vincenzo Librandi_ Mar 26 2019
%H A082652 Robert Israel, <a href="/A082652/b082652.txt">Table of n, a(n) for n = 1..10011</a>
%H A082652 Antonio Bernini, Matteo Cervetti, Luca Ferrari, and Einar Steingrimsson, <a href="https://arxiv.org/abs/1910.00299">Enumerative combinatorics of intervals in the Dyck pattern poset</a>, arXiv:1910.00299 [math.CO], 2019. See p. 5.
%F A082652 T(n, k) = ( k + 3*k*n + 3*k^2*n - k^3 ) / 6.
%F A082652 T(n, k) = Sum_{i=0..min(n,k)} (n-i)*(k-i). - _N. J. A. Sloane_, Nov 17 2007
%F A082652 G.f.: (1+x*y-2*x^2*y)*x*y/((1-x*y)^4*(1-x)^2). - _Robert Israel_, Dec 20 2017
%e A082652 Let X represent a small square. Then T(3,2) = 8 because here
%e A082652   XXX
%e A082652   XXX
%e A082652 we can see 8 squares, 6 of side 1, 2 of side 2.
%e A082652 Triangle begins:
%e A082652   1
%e A082652   2   5
%e A082652   3   8  14
%e A082652   4  11  20  30
%e A082652   5  14  26  40  55
%e A082652   6  17  32  50  70  91
%e A082652   7  20  38  60  85 112 140
%e A082652   ...
%p A082652 f:=proc(m,n) add((m-i)*(n-i),i=0..min(m,n)); end;
%t A082652 T[n_, k_] := Sum[(n-i)(k-i), {i, 0, Min[n, k]}];
%t A082652 Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Mar 25 2019 *)
%o A082652 (Magma) /* As triangle */ [[(k+3*k*n+3*k^2*n-k^3)/6: k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Mar 26 2019
%Y A082652 Cf. A083003, A083487. Right side of triangle gives A000330.
%Y A082652 Main diagonal is A000330, row sums are A001296. - _Paul D. Hanna_ and other correspondents, May 28 2003
%Y A082652 Cf. A130684. - _Joel B. Lewis_
%K A082652 nonn,tabl
%O A082652 1,2
%A A082652 Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), May 16 2003
%E A082652 Edited by _Robert Israel_, Dec 20 2017