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Triangle read by rows: T(n,k) (n>=0, 0 <= k <= n) = number of decreasing lines defining the Farey diagram Farey(n,k) of order (n,k).
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%I #15 Apr 19 2023 09:04:49

%S 0,0,2,0,5,10,0,9,19,32,0,14,27,47,66,0,20,40,68,96,134,0,27,51,85,

%T 118,167,204,0,35,68,112,156,217,267,342,0,44,82,137,187,261,318,408,

%U 482,0,54,103,166,229,317,384,490,581,692,0,65,120,196,266,366,441,564,664,794,904

%N Triangle read by rows: T(n,k) (n>=0, 0 <= k <= n) = number of decreasing lines defining the Farey diagram Farey(n,k) of order (n,k).

%H Daniel Khoshnoudirad, <a href="http://www.doiserbia.nb.rs/img/doi/1452-8630/2015/1452-86301500008K.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 The full array T(n,k), n >= 0, k>= 0, begins:

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, ...

%e 0, 5, 10, 19, 27, 40, 51, 68, 82, 103, 120, 145, ...

%e 0, 9, 19, 32, 47, 68, 85, 112, 137, 166, 196, 235, ...

%e 0, 14, 27, 47, 66, 96, 118, 156, 187, 229, 266, ...

%e 0, 20, 40, 68, 96, 134, 167, 217, 261, 317, 366, ...

%e 0, 27, 51, 85, 118, 167, 204, 267, 318, 384, 441, ...

%p A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper

%p Amn:=proc(m,n) local a,i,j; # A331781 or equally A333295. Diagonal is A018805.

%p a:=0; for i from 1 to m do for j from 1 to n do

%p if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end;

%p DFD:=proc(m,n) local d,t1,u,v; global A005728, Amn;

%p t1:=0; for u from 1 to m do for v from 1 to n do

%p d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:

%p t1; end;

%p for m from 0 to 8 do lprint([seq(DFD(m,n),n=0..20)]); od:

%t T[n_, k_] := Sum[d = GCD[u, v]; If[d >= 1, (u+v)*EulerPhi[d]/d, 0], {u, 1, n}, {v, 1, k}];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 18 2023 *)

%Y Cf. A358298.

%Y 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 nonn,tabl

%O 0,3

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 06 2022