A332353 - OEIS

%I #20 Oct 19 2023 11:28:24

%S 0,0,0,1,2,8,2,4,14,24,3,6,22,38,60,4,8,30,52,82,112,5,10,40,70,112,

%T 154,212,6,12,50,88,142,196,270,344,7,14,62,110,178,246,340,434,548,8,

%U 16,74,132,214,296,410,524,662,800,9,18,88,158,258,358,498,638,808,978,1196

%N Triangle read by rows: T(m,n) = Sum_{-m<i<m, -n<j<n, gcd{i,j}=2} (m-|i|)*(n-|j|)/2, m >= n >= 1.

%C This is the triangle in A332352, halved.

%H Paolo Xausa, <a href="/A332353/b332353.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of the triangle, flattened)

%H M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. <a href="https://doi.org/10.1137/140978090">On the minimal teaching sets of two-dimensional threshold functions</a>. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. This sequence is f_2(m,n)/2.

%e Triangle begins:

%e 0,

%e 0, 0,

%e 1, 2, 8,

%e 2, 4, 14, 24,

%e 3, 6, 22, 38, 60,

%e 4, 8, 30, 52, 82, 112,

%e 5, 10, 40, 70, 112, 154, 212,

%e 6, 12, 50, 88, 142, 196, 270, 344,

%e 7, 14, 62, 110, 178, 246, 340, 434, 548,

%e 8, 16, 74, 132, 214, 296, 410, 524, 662, 800,

%e ...

%p VR := proc(m,n,q) local a,i,j; a:=0;

%p for i from -m+1 to m-1 do for j from -n+1 to n-1 do

%p if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;

%p for m from 1 to 12 do lprint(seq(VR(m,n,2)/2,n=1..m),); od:

%t A332353[m_,n_]:=Sum[If[GCD[i,j]==2,2(m-i)(n-j),0],{i,2,m-1,2},{j,2,n-1,2}]+If[n>2,m*n-2m,0]+If[m>2,m*n-2n,0];Table[A332353[m, n],{m,15},{n, m}] (* _Paolo Xausa_, Oct 18 2023 *)

%Y The main diagonal is A177719.

%Y Cf. A332350, A332352.

%K nonn,tabl

%O 1,5

%A _N. J. A. Sloane_, Feb 10 2020