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A358304 Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of decreasing lines defining the Farey diagram Farey(n,k) of order (n,k).

%I A358304 #26 Apr 20 2023 02:30:32
%S A358304 0,0,0,0,2,0,0,5,5,0,0,9,10,9,0,0,14,19,19,14,0,0,20,27,32,27,20,0,0,
%T A358304 27,40,47,47,40,27,0,0,35,51,68,66,68,51,35,0,0,44,68,85,96,96,85,68,
%U A358304 44,0,0,54,82,112,118,134,118,112,82,54,0,0,65,103,137,156,167,167,156,137,103,65,0,0,77,120,166,187,217,204,217,187,166,120,77,0
%N A358304 Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of decreasing lines defining the Farey diagram Farey(n,k) of order (n,k).
%H A358304 Daniel Khoshnoudirad, <a href="http://pefmath.etf.rs/vol9num1/AADM-Vol9-No1-73-84.pdf">Farey lines defining Farey diagrams and application to some discrete structures</a>, Applicable Analysis and Discrete Mathematics, 9 (2015), 73-84; doi:10.2298/AADM150219008K. See Lemma 1, |DFD(m,n)|.
%e A358304 The full array T(n,k), n >= 0, k >= 0, begins:
%e A358304   0,  0,  0,  0,   0,   0,   0,   0,   0,   0,   0,   0,   0, ..
%e A358304   0,  2,  5,  9,  14,  20,  27,  35,  44,  54,  65,  77,  90, ..
%e A358304   0,  5, 10, 19,  27,  40,  51,  68,  82, 103, 120, 145, 165, ..
%e A358304   0,  9, 19, 32,  47,  68,  85, 112, 137, 166, 196, 235, 265, ..
%e A358304   0, 14, 27, 47,  66,  96, 118, 156, 187, 229, 266, 320, 358, ..
%e A358304   0, 20, 40, 68,  96, 134, 167, 217, 261, 317, 366, 436, 491, ..
%e A358304   0, 27, 51, 85, 118, 167, 204, 267, 318, 384, 441, 528, 589, ..
%e A358304   ...
%p A358304 A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
%p A358304 Amn:=proc(m,n) local a,i,j;  # A331781 or equally A333295. Diagonal is A018805.
%p A358304 a:=0; for i from 1 to m do for j from 1 to n do
%p A358304 if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
%p A358304 DFD:=proc(m,n) local d,t1,u,v; global A005728, Amn;
%p A358304 t1:=0; for u from 1 to m do for v from 1 to n do
%p A358304 d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:
%p A358304 t1; end;
%p A358304 for m from 0 to 8 do lprint([seq(DFD(m,n),n=0..20)]); od:
%t A358304 T[n_, k_] := Sum[d = GCD[u, v]; If[d >= 1, (u+v)*EulerPhi[d]/d, 0], {u, 1, n}, {v, 1, k}];
%t A358304 Table[T[n-k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 18 2023 *)
%Y A358304 Cf. A358298.
%Y A358304 The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.
%K A358304 nonn,tabl
%O A358304 0,5
%A A358304 _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 06 2022