

A082652


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


6



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

T(n,k) also is the total number of balls in a pyramid of balls on a 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).
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
Here the squares being counted have sides parallel to the gridlines; for all squares, see A130684.


LINKS

Robert Israel, Table of n, a(n) for n = 1..10011


FORMULA

T(n, k) = ( k + 3*k*n + 3*k^2*n  k^3 ) / 6.
T(n, k) = Sum{ i = 0..min(n,k)} (ni)*(ki).  N. J. A. Sloane, Nov 17 2007
G.f.: (1+x*y2*x^2*y)*x*y/((1x*y)^4*(1x)^2).  Robert Israel, Dec 20 2017


EXAMPLE

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.


MAPLE

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


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
Sequence in context: A262870 A294210 A244418 * A194007 A065222 A159988
Adjacent sequences: A082649 A082650 A082651 * A082653 A082654 A082655


KEYWORD

nonn,tabl


AUTHOR

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


EXTENSIONS

Edited by Robert Israel, Dec 20 2017


STATUS

approved



