A144000 Rectangular array by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares for which x + y == 0 (mod 3); then R(m,n) is the number of marked squares in the rectangle [0,m]x[0,n].

1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 3, 5, 6, 5, 3, 4, 6, 8, 8, 6, 4, 5, 8, 10, 11, 10, 8, 5, 5, 9, 12, 13, 13, 12, 9, 5, 6, 10, 14, 16, 16, 16, 14, 10, 6, 7, 12, 16, 19, 20, 20, 19, 16, 12, 7, 7, 13, 18, 21, 23, 24, 23, 21, 18, 13, 7, 8, 14, 20, 24, 26, 28, 28, 26, 24, 20, 14, 8, 9, 16, 22, 27, 30

Offset: 1

Author

Clark Kimberling, Sep 07 2008

Keywords

• nonn
• tabl

Comments

Row 3n is given by 2n*(1,2,3,4,5,6,...).

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A143996, A143997, A143998, A144399, A144001.

Programs

Maple

A := proc(n,k) ## n = 0 .. infinity and k = 0 .. n
    if  1 = (n-k+1) mod 3 then
        floor((2*(k+1)*(n-k+1)+1) / 3)
    else
        floor((2*(k+1)*(n-k+1)) / 3)
    end if
end proc: # Yu-Sheng Chang, Jan 01 2020

Mathematica

b[n_, m_] := If[Mod[n, 3] == 1, Floor[(2*m*n + 1)/3],  Floor[2*m*n/3]]; a:= Table[a[n, m], {n, 1, 25}, {m, 1, 25}]; Table[a[[k, n - k + 1]], {n, 1, 20}, {k, 1, n}]//Flatten (* G. C. Greubel, Dec 05 2017 *)

Formula

R(m,n) = floor((2*m*n + 1)/3) if n == 1 (mod 3) and floor(2*m*n/3) otherwise.