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.

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, 112, 140, 8, 23, 44, 70, 100, 133, 168, 204, 9, 26, 50, 80, 115, 154, 196, 240, 285, 10, 29, 56, 90, 130, 175, 224, 276, 330, 385, 11, 32, 62, 100, 145, 196, 252, 312, 375, 440, 506

Offset: 1

Author

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), May 16 2003

Keywords

• nonn
• tabl

Comments

Here the squares being counted have sides parallel to the gridlines; for all squares, see A130684.

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).

Row sums give A001296. - Vincenzo Librandi Mar 26 2019

Examples

			Let X represent a small square. Then T(3,2) = 8 because here
  XXX
  XXX
we can see 8 squares, 6 of side 1, 2 of side 2.
Triangle begins:
  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  85 112 140
  ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10011
• Antonio Bernini, Matteo Cervetti, Luca Ferrari, and Einar Steingrimsson, Enumerative combinatorics of intervals in the Dyck pattern poset, arXiv:1910.00299 [math.CO], 2019. See p. 5.

Crossrefs

Cf. A083003, A083487. Right side of triangle gives A000330.

Main diagonal is A000330, row sums are A001296. - Paul D. Hanna and other correspondents, May 28 2003

Cf. A130684. - Joel B. Lewis

Programs

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

Maple

f:=proc(m,n) add((m-i)*(n-i),i=0..min(m,n)); end;

Mathematica

T[n_, k_] := Sum[(n-i)(k-i), {i, 0, Min[n, k]}];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2019 *)

Formula

T(n, k) = ( k + 3*k*n + 3*k^2*n - k^3 ) / 6.

T(n, k) = Sum_{i=0..min(n,k)} (n-i)*(k-i). - N. J. A. Sloane, Nov 17 2007

G.f.: (1+x*y-2*x^2*y)*x*y/((1-x*y)^4*(1-x)^2). - Robert Israel, Dec 20 2017

Extensions

Edited by Robert Israel, Dec 20 2017