

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
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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 selffusion 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*(m1)*(3*nm1)/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/((1x)^2*(1x*y)^2*(1x^2*y)).
Number triangle T(n, k) = sum{j=0..n, [j<=k]*(kj+1)[j<=nk]*(nkj+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



