A115009 Array read by antidiagonals: let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)*(n+1-j), then T(m,n) = 2*m*n+m+n+2*V(m,n), for m >= 0, n >= 0.

0, 1, 1, 2, 6, 2, 3, 13, 13, 3, 4, 22, 28, 22, 4, 5, 33, 49, 49, 33, 5, 6, 46, 74, 86, 74, 46, 6, 7, 61, 105, 131, 131, 105, 61, 7, 8, 78, 140, 188, 200, 188, 140, 78, 8, 9, 97, 181, 251, 289, 289, 251, 181, 97, 9, 10, 118, 226, 326, 386, 418, 386, 326, 226, 118, 10, 11, 141, 277

Offset

0,4

Comments

This is the number of linear partitions of an m X n grid.

References

D. M. Acketa, J. D. Zunic: On the number of linear partitions of the (m,n)-grid. Inform. Process. Lett., 38 (3) (1991), 163-168. See Table A.1.

Jovisa Zunic, Note on the number of two-dimensional threshold functions, SIAM J. Discrete Math. Vol. 25 (2011), No. 3, pp. 1266-1268. See Equation (1.2).

Links

Table of n, a(n) for n=0..68.

Max A. Alekseyev. On the number of two-dimensional threshold functions. SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631. doi:10.1137/090750184

Example

The array begins:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
1, 6, 13, 22, 33, 46, 61, 78, 97, 118, ...
2, 13, 28, 49, 74, 105, 140, 181, 226, 277, ...
3, 22, 49, 86, 131, 188, 251, 326, 409, 502, ...
4, 33, 74, 131, 200, 289, 386, 503, 632, 777, ...
5, 46, 105, 188, 289, 418, 559, 730, 919, 1132, ...
6, 61, 140, 251, 386, 559, 748, 979, 1234, 1521, ...
7, 78, 181, 326, 503, 730, 979, 1282, 1617, 1994, ...
...

Maple

V:=proc(m, n) local t1, i, j; t1:=0; for i from 1 to m do for j from 1 to n do if gcd(i, j)=1 then t1:=t1+(m+1-i)*(n+1-j); fi; od; od; t1; end; T:=(m, n)->(2*m*n+m+n+2*V(m, n));

Mathematica

V[m_, n_] := Sum[If[GCD[i, j] == 1, (m-i+1)*(n-j+1), 0], {i, 1, m}, {j, 1, n}]; T[m_, n_] := 2*m*n+m+n+2*V[m, n]; Table[T[m-n, n], {m, 0, 11}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

Crossrefs

The second and third rows are A028872 and A358296.

The main diagonal is A141255 = A114043 - 1.

The lower triangle is A332351.

Cf. A114999, A114043, A115004, A115005, A115006, A115007, A115010, A115011.

Sequence in context: A057892 A334188 A265993 * A378932 A151944 A073094

Adjacent sequences: A115006 A115007 A115008 * A115010 A115011 A115012

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 24 2006

Status

approved