A115262 Correlation triangle for n+1.

1, 2, 2, 3, 5, 3, 4, 8, 8, 4, 5, 11, 14, 11, 5, 6, 14, 20, 20, 14, 6, 7, 17, 26, 30, 26, 17, 7, 8, 20, 32, 40, 40, 32, 20, 8, 9, 23, 38, 50, 55, 50, 38, 23, 9, 10, 26, 44, 60, 70, 70, 60, 44, 26, 10, 11, 29, 50, 70, 85, 91, 85, 70, 50, 29, 11

Offset: 0

Author

Paul Barry, Jan 18 2006

Keywords

• easy
• nonn
• tabl

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.

The accumulation array of A115262 is A185957. (End)

From Clark Kimberling, Dec 22 2011: (Start)

As a square matrix, A115262 is the self-fusion matrix of A000027 (1,2,3,4,...). See A193722 for the definition of fusion and A202673 for characteristic polynomials associated with A115622. (End)

Examples

			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, ...
  ...
		

Crossrefs

Cf. A000027, A202673, A271916.

For the triangular version: row sums are A001752. Diagonal sums are A097701. T(2n,n) is A000330(n+1).

Diagonals (1,5,...): A000330 (square pyramidal numbers),

diagonals (2,8,...): A007290,

diagonals (3,11,...): A051925,

diagonals (4,14,...): A159920,

antidiagonal sums: A001752.

Programs

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 *)

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.