|
|
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.
|
|
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, 506
(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 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).
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
|
|
|
FORMULA
|
T(n, k) = ( k + 3*k*n + 3*k^2*n - k^3 ) / 6.
G.f.: (1+x*y-2*x^2*y)*x*y/((1-x*y)^4*(1-x)^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((m-i)*(n-i), i=0..min(m, n)); end;
|
|
MATHEMATICA
|
T[n_, k_] := Sum[(n-i)(k-i), {i, 0, Min[n, k]}];
|
|
PROG
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), May 16 2003
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|