OFFSET
0,5
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)|.
EXAMPLE
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, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, ..
0, 5, 10, 19, 27, 40, 51, 68, 82, 103, 120, 145, 165, ..
0, 9, 19, 32, 47, 68, 85, 112, 137, 166, 196, 235, 265, ..
0, 14, 27, 47, 66, 96, 118, 156, 187, 229, 266, 320, 358, ..
0, 20, 40, 68, 96, 134, 167, 217, 261, 317, 366, 436, 491, ..
0, 27, 51, 85, 118, 167, 204, 267, 318, 384, 441, 528, 589, ..
...
MAPLE
A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
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:
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, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 18 2023 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Scott R. Shannon and N. J. A. Sloane, Dec 06 2022
STATUS
approved