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A115262 Correlation triangle for n+1. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence (formatted as a square array) gives the counts of all possible squares in an mXn rectangle. For example, 11 = 8 (1X1 square)+ 3 (2X2 square) in 4X2 rectangle. [From 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)

LINKS

Table of n, a(n) for n=0..65.

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}]]

(* from Clark Kimberling, Dec 22 2011 *)

CROSSREFS

Cf. A000027, A202673, A271916.

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

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

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

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

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

antidiagonal sums: A001752.

Sequence in context: A132071 A061177 A129312 * A128141 A252829 A014430

Adjacent sequences:  A115259 A115260 A115261 * A115263 A115264 A115265

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Jan 18 2006

STATUS

approved

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Last modified November 15 21:17 EST 2019. Contains 329151 sequences. (Running on oeis4.)