A358302 Number of triangular regions in the Farey Diagram Farey(n,n), divided by 4.
1, 12, 100, 392, 1554, 3486, 9690, 18942, 38610, 65268, 125116, 186870, 324646, 472546, 713354, 1003888, 1531908, 2000638, 2920970, 3780950
Offset: 1
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The full array T(n,k), n >= 0, k>= 0, begins: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, ... 0, 5, 10, 19, 27, 40, 51, 68, 82, 103, 120, 145, ... 0, 9, 19, 32, 47, 68, 85, 112, 137, 166, 196, 235, ... 0, 14, 27, 47, 66, 96, 118, 156, 187, 229, 266, ... 0, 20, 40, 68, 96, 134, 167, 217, 261, 317, 366, ... 0, 27, 51, 85, 118, 167, 204, 267, 318, 384, 441, ...
A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper Amn:=proc(m,n) local a,i,j; # A331781 or equally A333295. Diagonal is A018805. a:=0; for i from 1 to m do for j from 1 to n do if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end; DFD:=proc(m,n) local d,t1,u,v; global A005728, Amn; t1:=0; for u from 1 to m do for v from 1 to n do d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od: t1; end; for m from 0 to 8 do lprint([seq(DFD(m,n),n=0..20)]); od:
T[n_, k_] := Sum[d = GCD[u, v]; If[d >= 1, (u+v)*EulerPhi[d]/d, 0], {u, 1, n}, {v, 1, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 18 2023 *)
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