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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.
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%I #23 Dec 05 2022 20:50:56

%S 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,

%T 6,7,61,105,131,131,105,61,7,8,78,140,188,200,188,140,78,8,9,97,181,

%U 251,289,289,251,181,97,9,10,118,226,326,386,418,386,326,226,118,10,11,141,277

%N 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.

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

%D 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.

%D 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).

%H Max A. Alekseyev. <a href="http://arXiv.org/abs/math.CO/0602511">On the number of two-dimensional threshold functions</a>. SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631. doi:10.1137/090750184

%e The array begins:

%e 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...

%e 1, 6, 13, 22, 33, 46, 61, 78, 97, 118, ...

%e 2, 13, 28, 49, 74, 105, 140, 181, 226, 277, ...

%e 3, 22, 49, 86, 131, 188, 251, 326, 409, 502, ...

%e 4, 33, 74, 131, 200, 289, 386, 503, 632, 777, ...

%e 5, 46, 105, 188, 289, 418, 559, 730, 919, 1132, ...

%e 6, 61, 140, 251, 386, 559, 748, 979, 1234, 1521, ...

%e 7, 78, 181, 326, 503, 730, 979, 1282, 1617, 1994, ...

%e ...

%p 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));

%t 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 *)

%Y The second and third rows are A028872 and A358296.

%Y The main diagonal is A141255 = A114043 - 1.

%Y The lower triangle is A332351.

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

%K nonn,tabl

%O 0,4

%A _N. J. A. Sloane_, Feb 24 2006