OFFSET
0,2
COMMENTS
This sequence (formatted as a square array) gives the counts of all possible squares in an m X n rectangle. For example, 11 = 8 (1 X 1 squares) + 3 (2 X 2 square) in 4 X 2 rectangle. - Philippe Deléham, Nov 26 2009
From Clark Kimberling, Feb 07 2011: (Start)
Also the accumulation array of min{n,k}, when formatted as a rectangle.
This is the accumulation array of the array M=A003783 given by M(n,k)=min{n,k}; see A144112 for the definition of accumulation array.
From Clark Kimberling, Dec 22 2011: (Start)
FORMULA
Let f(m,n) = m*(m-1)*(3*n-m-1)/6. This array is (with a different offset) the infinite square array read by antidiagonals U(m,n) = f(n,m) if m < n, U(m,n) = f(m,n) if m <= n. See A271916. - N. J. A. Sloane, Apr 26 2016
G.f.: 1/((1-x)^2*(1-x*y)^2*(1-x^2*y)).
Number triangle T(n, k) = Sum_{j=0..n} [j<=k]*(k-j+1)[j<=n-k]*(n-k-j+1).
T(2n,n) - T(2n,n+1) = n+1.
EXAMPLE
Triangle begins
1;
2, 2;
3, 5, 3;
4, 8, 8, 4;
5, 11, 14, 11, 5;
6, 14, 20, 20, 14, 6;
...
When formatted as a square matrix:
1, 2, 3, 4, 5, ...
2, 5, 8, 11, 14, ...
3, 8, 14, 20, 26, ...
4, 11, 20, 30, 40, ...
5, 14, 26, 40, 55, ...
...
MATHEMATICA
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[k, {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
(* Clark Kimberling, Dec 22 2011 *)
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jan 18 2006
STATUS
approved