A358305 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).
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, 118, 167, 204, 0, 35, 68, 112, 156, 217, 267, 342, 0, 44, 82, 137, 187, 261, 318, 408, 482, 0, 54, 103, 166, 229, 317, 384, 490, 581, 692, 0, 65, 120, 196, 266, 366, 441, 564, 664, 794, 904
Offset: 0
Examples
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, ...
Links
- Daniel Khoshnoudirad, Farey lines defining Farey diagrams and application to some discrete structures, Applicable Analysis and Discrete Mathematics, 9 (2015), 73-84; doi:10.2298/AADM150219008K. See Lemma 1, |DFD(m,n)|.
Crossrefs
Programs
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Maple
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:
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Mathematica
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 *)